Class 10th Mathematics Tamilnadu Board Solution
Exercise 5.1- Find the midpoint of the line segment joining the points (i) (1, - 1) and (-5,…
- Find the centroid of the triangle whose vertices are (i) (1,3), (2, 7) and (12,…
- The center of a circle is at (-6, 4). If one end of a diameter of the circle is…
- If the centroid of a triangle is at (1, 3) and two of its vertices are (-7, 6)…
- Using the section formula, show that the points A (1,0), B (5,3), C (2,7) and D…
- Find the coordinates of the point which divides the line segment joining (3, 4)…
- Find the coordinates of the point which divides the line segment joining (-3, 5)…
- Let A (-6, -5) and B (-6, 4) be two points such that a point P on the line AB…
- Find the points of trisection of the line segment joining the points A (2, -2)…
- Find the points which divide the line segment joining A (-4 ,0) and B (0,6)…
- Find the ratio in which the x-axis divides the line segment joining the points…
- In what ratio is the line joining the points (-5, 1) and (2, 3) divided by the…
- Find the length of the medians of the triangle whose vertices are (1, -1), (0,…
Exercise 5.2- (0, 0), (3, 0) and (0, 2) Find the area of the triangle formed by the points…
- (5, 2), (3, -5) and (-5, -1) Find the area of the triangle formed by the points…
- (-4, -5), (4, 5) and (-1, -6) Find the area of the triangle formed by the…
- Vertices: (0, 0), (4, a), (6, 4) Area (in sq. units): 17 Vertices of the…
- Vertices: (a, a), (4, 5), (6, -1) Area (in sq. units): 9 Vertices of the…
- Vertices: (a, -3), (3, a), (-1,5) Area (in sq. units): 12 Vertices of the…
- (4, 3), (1, 2) and (-2, 1) Determine if the following set of points are…
- (-2, -2), (-6, -2) and (-2, 2) Determine if the following set of points are…
- (- 3/2 , 3) , (6, -2) and (-3, 4) Determine if the following set of points are…
- (k, -1), (2, 1) and (4, 5) In each of the following, find the value of k for…
- (2, - 5), (3, - 4) and (9, k) In each of the following, find the value of k for…
- (k, k), (2, 3) and (4, - 1) In each of the following, find the value of k for…
- (6, 9), (7, 4), (4,2) and (3,7) Find the area of the quadrilateral whose…
- (-3, 4), (-5, - 6), (4, - 1) and (1, 2) Find the area of the quadrilateral…
- (-4, 5), (0, 7), (5, - 5) and (-4, - 2) Find the area of the quadrilateral…
- If the three points (h, 0), (a, b) and (0, k) lie on a straight line, then using…
- Find the area of the triangle formed by joining the midpoints of the sides of a…
Exercise 5.3- Find the angle of inclination of the straight line whose slope is (i) 1 (ii) √3…
- Find the slope of the straight line whose angle of inclination is (i) 30° (ii)…
- Find the slope of the straight line passing through the points (i) (3, -2) and…
- Find the angle of inclination of the line passing through the points (i) (1, 2)…
- Find the slope of the line which passes through the origin and the midpoint of…
- The side AB of a square ABCD is parallel to x-axis. Find the (i) slope of AB…
- The side BC of an equilateral 3ABC is parallel to x-axis. Find the slope of AB…
- (2, 3), (3, -1) and (4, -5) Using the concept of slope, show that each of the…
- (4, 1), (-2, -3) and (-5, -5) Using the concept of slope, show that each of the…
- (4, 4), (-2, 6) and (1, 5) Using the concept of slope, show that each of the…
- If the points (a, 1), (1, 2) and (0, b + 1) are collinear, then show that 1/a +…
- The line joining the points A (-2, 3) and B (a, 5) is parallel to the line…
- The line joining the points A (0, 5) and B (4, 2) is perpendicular to the line…
- The vertices of ΔABC are A (1, 8), B (-2, 4), C (8, -5). If M and N are the…
- A triangle has vertices at (6, 7), (2, -9) and (-4, 1). Find the slopes of its…
- The vertices of a ΔABC are A (-5, 7), B (-4, -5) and C (4, 5). Find the slopes…
- Using the concept of slope, show that the vertices (1, 2), (-2, 2), (-4, -3)…
- Show that the opposite sides of a quadrilateral with vertices A (-2, -4), B (5,…
Exercise 5.4- Write the equations of the straight lines parallel to x- axis which are at a…
- Find the equations of the straight lines parallel to the coordinate axes and…
- Find the equation of a straight line whose (i) slope is -3 and y-intercept is 4.…
- Find the equation of the line intersecting the y- axis at a distance of 3 units…
- Find the slope and y-intercept of the line whose equation is (i) y = x + 1 (ii)…
- Find the equation of the straight line whose (i) slope is -4 and passing through…
- Find the equation of the straight line which passes through the midpoint of the…
- Find the equation of the straight line passing through the points (i) (-2, 5)…
- Find the equation of the median from the vertex R in a ΔPQR with vertices at…
- By using the concept of the equation of the straight line, prove that the given…
- Find the equation of the straight line whose x and y-intercepts on the axes are…
- Find the x and y intercepts of the straight line (i) 5x + 3y - 15 = 0 (ii) 2x -…
- Find the equation of the straight line passing through the point (3, 4) and has…
- Find the equation of the straight lines passing through the point (2, 2) and…
- Find the equation of the straight line passing through the point (5, -3) and…
- Find the equation of the line passing through the point (9, -1) and having its…
- A straight line cuts the coordinate axes at A and B. If the midpoint of AB is…
- Find the equation of the line passing through (22, -6) and having intercept on…
- If A(3, 6) and C(-1, 2) are two vertices of a rhombus ABCD, then find the…
- Find the equation of the line whose gradient is 3/2 and which passes through P,…
Exercise 5.5- 3x + 4y - 6 = 0 Find the slope of the straight line
- y = 7x + 6 Find the slope of the straight line
- 4x = 5y + 3. Find the slope of the straight line
- Show that the straight lines x + 2y + 1 = 0 and 3x + 6y + 2 = 0 are parallel.…
- Show that the straight lines 3x - 5y + 7 = 0 and 15x + 9y + 4 = 0 are…
- If the straight lines y/2 = x-p and ax + 5 = 3y are parallel, then find a.…
- Find the value of a if the straight lines 5x - 2y - 9 = 0 and ay + 2x - 11 = 0…
- Find the values of p for which the straight lines 8px + (2 - 3p)y + 1 = 0 and px…
- If the straight line passing through the points (h, 3)and (4, 1) intersects the…
- Find the equation of the straight line parallel to the line 3x - y + 7 = 0 and…
- Find the equation of the straight line perpendicular to the straight line x - 2y…
- Find the equation of the perpendicular bisector of the straight line segment…
- Find the equation of the straight line passing through the point of…
- Find the equation of the straight line which passes through the point of…
- Find the equation of the straight line joining the point of intersection of the…
- If the vertices of a Δ ABC are A(2, -4), B(3, 3) and C(-1, 5). Find the…
- If the vertices of a Δ ABC are A(-4,4), B(8 ,4) and C(8,10). Find the equation…
- Find the coordinates of the foot of the perpendicular from the origin on the…
- If x + 2y = 7 and 2x + y = 8 are the equations of the lines of two diameters of…
- Find the equation of the straight line segment whose end points are the point…
- In an isosceles Δ PQR, PQ = PR. The base QR lies on the x-axis, P lies on the…
Exercise 5.6- The midpoint of the line joining (a,- b) and (3a, 5b) isA. (-a, 2b) B. (2a, 4b)…
- The point P which divides the line segment joining the points A(1,- 3)and B(-3,…
- If the line segment joining the points A(3, 4) and B (14,- 3)meets the x-axis at…
- The centroid of the triangle with vertices at (-2,- 5), (-2,12) and (10, -…
- If (1, 2), (4, 6), (x, 6)and (3, 2)are the vertices of a parallelogram taken in…
- Area of the triangle formed by the points (0,0), (2, 0)and (0, 2)isA. 1 sq.…
- Area of the quadrilateral formed by the points (1,1), (0,1), (0, 0)and (1,…
- The angle of inclination of a straight line parallel to x-axis is equal toA. 0°…
- Slope of the line joining the points (3,- 2)and (-1, a) is - 3/2 , then the…
- Slope of the straight line which is perpendicular to the straight line joining…
- The point of intersection of the straight lines 9x - y - 2 = 0 and 2x + y - 9 =…
- The straight line 4x + 3y - 12 = 0 intersects the y- axis atA. (3, 0) B. (0, 4)…
- The slope of the straight line 7y - 2x = 11 is equal toA. - 7/2 B. 7/2 C. 2/7…
- The equation of a straight line passing through the point (2 , -7) and parallel…
- The x and y-intercepts of the line 2x - 3y + 6 = 0, respectively areA. 2, 3 B.…
- The centre of a circle is (-6, 4). If one end of the diameter of the circle is…
- The equation of the straight line passing through the origin and perpendicular…
- The equation of a straight line parallel to y-axis and passing through the…
- If the points (2, 5), (4, 6) and (a, a) are collinear, then the value of a is…
- If a straight line y = 2x + k passes through the point (1, 2), then the value…
- The equation of a straight line having slope 3 and y-intercept -4 isA. 3x - y -…
- The point of intersection of the straight lines y = 0 and x = - 4 isA. (0,- 4)…
- The value of k if the straight lines 3x + 6y + 7 = 0 and 2x + ky = 5 are…
- Find the midpoint of the line segment joining the points (i) (1, - 1) and (-5,…
- Find the centroid of the triangle whose vertices are (i) (1,3), (2, 7) and (12,…
- The center of a circle is at (-6, 4). If one end of a diameter of the circle is…
- If the centroid of a triangle is at (1, 3) and two of its vertices are (-7, 6)…
- Using the section formula, show that the points A (1,0), B (5,3), C (2,7) and D…
- Find the coordinates of the point which divides the line segment joining (3, 4)…
- Find the coordinates of the point which divides the line segment joining (-3, 5)…
- Let A (-6, -5) and B (-6, 4) be two points such that a point P on the line AB…
- Find the points of trisection of the line segment joining the points A (2, -2)…
- Find the points which divide the line segment joining A (-4 ,0) and B (0,6)…
- Find the ratio in which the x-axis divides the line segment joining the points…
- In what ratio is the line joining the points (-5, 1) and (2, 3) divided by the…
- Find the length of the medians of the triangle whose vertices are (1, -1), (0,…
- (0, 0), (3, 0) and (0, 2) Find the area of the triangle formed by the points…
- (5, 2), (3, -5) and (-5, -1) Find the area of the triangle formed by the points…
- (-4, -5), (4, 5) and (-1, -6) Find the area of the triangle formed by the…
- Vertices: (0, 0), (4, a), (6, 4) Area (in sq. units): 17 Vertices of the…
- Vertices: (a, a), (4, 5), (6, -1) Area (in sq. units): 9 Vertices of the…
- Vertices: (a, -3), (3, a), (-1,5) Area (in sq. units): 12 Vertices of the…
- (4, 3), (1, 2) and (-2, 1) Determine if the following set of points are…
- (-2, -2), (-6, -2) and (-2, 2) Determine if the following set of points are…
- (- 3/2 , 3) , (6, -2) and (-3, 4) Determine if the following set of points are…
- (k, -1), (2, 1) and (4, 5) In each of the following, find the value of k for…
- (2, - 5), (3, - 4) and (9, k) In each of the following, find the value of k for…
- (k, k), (2, 3) and (4, - 1) In each of the following, find the value of k for…
- (6, 9), (7, 4), (4,2) and (3,7) Find the area of the quadrilateral whose…
- (-3, 4), (-5, - 6), (4, - 1) and (1, 2) Find the area of the quadrilateral…
- (-4, 5), (0, 7), (5, - 5) and (-4, - 2) Find the area of the quadrilateral…
- If the three points (h, 0), (a, b) and (0, k) lie on a straight line, then using…
- Find the area of the triangle formed by joining the midpoints of the sides of a…
- Find the angle of inclination of the straight line whose slope is (i) 1 (ii) √3…
- Find the slope of the straight line whose angle of inclination is (i) 30° (ii)…
- Find the slope of the straight line passing through the points (i) (3, -2) and…
- Find the angle of inclination of the line passing through the points (i) (1, 2)…
- Find the slope of the line which passes through the origin and the midpoint of…
- The side AB of a square ABCD is parallel to x-axis. Find the (i) slope of AB…
- The side BC of an equilateral 3ABC is parallel to x-axis. Find the slope of AB…
- (2, 3), (3, -1) and (4, -5) Using the concept of slope, show that each of the…
- (4, 1), (-2, -3) and (-5, -5) Using the concept of slope, show that each of the…
- (4, 4), (-2, 6) and (1, 5) Using the concept of slope, show that each of the…
- If the points (a, 1), (1, 2) and (0, b + 1) are collinear, then show that 1/a +…
- The line joining the points A (-2, 3) and B (a, 5) is parallel to the line…
- The line joining the points A (0, 5) and B (4, 2) is perpendicular to the line…
- The vertices of ΔABC are A (1, 8), B (-2, 4), C (8, -5). If M and N are the…
- A triangle has vertices at (6, 7), (2, -9) and (-4, 1). Find the slopes of its…
- The vertices of a ΔABC are A (-5, 7), B (-4, -5) and C (4, 5). Find the slopes…
- Using the concept of slope, show that the vertices (1, 2), (-2, 2), (-4, -3)…
- Show that the opposite sides of a quadrilateral with vertices A (-2, -4), B (5,…
- Write the equations of the straight lines parallel to x- axis which are at a…
- Find the equations of the straight lines parallel to the coordinate axes and…
- Find the equation of a straight line whose (i) slope is -3 and y-intercept is 4.…
- Find the equation of the line intersecting the y- axis at a distance of 3 units…
- Find the slope and y-intercept of the line whose equation is (i) y = x + 1 (ii)…
- Find the equation of the straight line whose (i) slope is -4 and passing through…
- Find the equation of the straight line which passes through the midpoint of the…
- Find the equation of the straight line passing through the points (i) (-2, 5)…
- Find the equation of the median from the vertex R in a ΔPQR with vertices at…
- By using the concept of the equation of the straight line, prove that the given…
- Find the equation of the straight line whose x and y-intercepts on the axes are…
- Find the x and y intercepts of the straight line (i) 5x + 3y - 15 = 0 (ii) 2x -…
- Find the equation of the straight line passing through the point (3, 4) and has…
- Find the equation of the straight lines passing through the point (2, 2) and…
- Find the equation of the straight line passing through the point (5, -3) and…
- Find the equation of the line passing through the point (9, -1) and having its…
- A straight line cuts the coordinate axes at A and B. If the midpoint of AB is…
- Find the equation of the line passing through (22, -6) and having intercept on…
- If A(3, 6) and C(-1, 2) are two vertices of a rhombus ABCD, then find the…
- Find the equation of the line whose gradient is 3/2 and which passes through P,…
- 3x + 4y - 6 = 0 Find the slope of the straight line
- y = 7x + 6 Find the slope of the straight line
- 4x = 5y + 3. Find the slope of the straight line
- Show that the straight lines x + 2y + 1 = 0 and 3x + 6y + 2 = 0 are parallel.…
- Show that the straight lines 3x - 5y + 7 = 0 and 15x + 9y + 4 = 0 are…
- If the straight lines y/2 = x-p and ax + 5 = 3y are parallel, then find a.…
- Find the value of a if the straight lines 5x - 2y - 9 = 0 and ay + 2x - 11 = 0…
- Find the values of p for which the straight lines 8px + (2 - 3p)y + 1 = 0 and px…
- If the straight line passing through the points (h, 3)and (4, 1) intersects the…
- Find the equation of the straight line parallel to the line 3x - y + 7 = 0 and…
- Find the equation of the straight line perpendicular to the straight line x - 2y…
- Find the equation of the perpendicular bisector of the straight line segment…
- Find the equation of the straight line passing through the point of…
- Find the equation of the straight line which passes through the point of…
- Find the equation of the straight line joining the point of intersection of the…
- If the vertices of a Δ ABC are A(2, -4), B(3, 3) and C(-1, 5). Find the…
- If the vertices of a Δ ABC are A(-4,4), B(8 ,4) and C(8,10). Find the equation…
- Find the coordinates of the foot of the perpendicular from the origin on the…
- If x + 2y = 7 and 2x + y = 8 are the equations of the lines of two diameters of…
- Find the equation of the straight line segment whose end points are the point…
- In an isosceles Δ PQR, PQ = PR. The base QR lies on the x-axis, P lies on the…
- The midpoint of the line joining (a,- b) and (3a, 5b) isA. (-a, 2b) B. (2a, 4b)…
- The point P which divides the line segment joining the points A(1,- 3)and B(-3,…
- If the line segment joining the points A(3, 4) and B (14,- 3)meets the x-axis at…
- The centroid of the triangle with vertices at (-2,- 5), (-2,12) and (10, -…
- If (1, 2), (4, 6), (x, 6)and (3, 2)are the vertices of a parallelogram taken in…
- Area of the triangle formed by the points (0,0), (2, 0)and (0, 2)isA. 1 sq.…
- Area of the quadrilateral formed by the points (1,1), (0,1), (0, 0)and (1,…
- The angle of inclination of a straight line parallel to x-axis is equal toA. 0°…
- Slope of the line joining the points (3,- 2)and (-1, a) is - 3/2 , then the…
- Slope of the straight line which is perpendicular to the straight line joining…
- The point of intersection of the straight lines 9x - y - 2 = 0 and 2x + y - 9 =…
- The straight line 4x + 3y - 12 = 0 intersects the y- axis atA. (3, 0) B. (0, 4)…
- The slope of the straight line 7y - 2x = 11 is equal toA. - 7/2 B. 7/2 C. 2/7…
- The equation of a straight line passing through the point (2 , -7) and parallel…
- The x and y-intercepts of the line 2x - 3y + 6 = 0, respectively areA. 2, 3 B.…
- The centre of a circle is (-6, 4). If one end of the diameter of the circle is…
- The equation of the straight line passing through the origin and perpendicular…
- The equation of a straight line parallel to y-axis and passing through the…
- If the points (2, 5), (4, 6) and (a, a) are collinear, then the value of a is…
- If a straight line y = 2x + k passes through the point (1, 2), then the value…
- The equation of a straight line having slope 3 and y-intercept -4 isA. 3x - y -…
- The point of intersection of the straight lines y = 0 and x = - 4 isA. (0,- 4)…
- The value of k if the straight lines 3x + 6y + 7 = 0 and 2x + ky = 5 are…
Exercise 5.1
Question 1.Find the midpoint of the line segment joining the points
(i) (1, – 1) and (–5, 3) (ii) (0,0) and (0,4)
Answer:(i). Midpoint of line–segment joining the points (x1, y1) and (x2, y2)
M (x, y) = M![](data:image/png;base64,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)
Midpoint of line–segment joining the points (1, –1) and (–5, 3)
M (x, y) = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= (–2, 1)
(ii). Midpoint of line–segment joining the points (x1, y1) and (x2, y2)
M (x, y) = M![](data:image/png;base64,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)
Midpoint of line–segment joining the points (0, 0) and (0, 4)
M (x, y) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEsAAAAjCAMAAADrNwfvAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOgA6OjoAOjo6OjpmOma2OpDbZgAAZgA6ZjoAZjpmZpDbZrbbZrb/kDoAkDo6kGY6kGZmkGaQkLbbkNv/tmYAtmY6trbbttv/tv//25A627Zm27a229uQ29u22////7Zm/9uQ/9u2//+2///bcPNkKgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABmElEQVRIS91W21bDIBDcRFsvNa2mqbeqpWpJw/9/oLC7EALmYpOHHvelnO0wzA4LBOCs4/j0cbo+uTh4k8ub19OpAMTtzk2vsrUeq8ckffApw0QEACjnZiaIS6dsO8PE7FBeoT5VEMIlaIUIAGpzjUhV3LEIYjDiOEdcXgKBEUCz57QqyAuukmRVS81IQ+aqE8QVAkAuuAInjGu20CpLMNbe3AZXDbjfqSJ/RqsECeOfv9cocE2ynQVteRP0b9P7OkHGRgC3S8ZM436VoUd6sErSHEfsQp1oAQB8zdl0VRhF5dzuJ804MbaGVCa0rSNDpLoNpuRCwvEhTHkTcWF5los6NA4nuRuAXFP6NSUXd/9Y97G/BvSqKvhodCyIJ3EAcDjEnu32RekO7wyuTgbN+r1KkkVj4hsdeRsxQHcDkdB14aJavsC+6yz8CrDFea8IM/I91l5VCHAuxHaIvmsoBNRyQmH7pl2xvBDgq6Hnx4boowoB9fOoKdTG+ySQmkp2NUEI0LMb0j/dFwE9WR1cEeD43td9/+f/H73LKEehAqS/AAAAAElFTkSuQmCC)
= ![](data:image/png;base64,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)
= (0, 2)
Question 2.Find the centroid of the triangle whose vertices are
(i) (1,3), (2, 7) and (12, – 16)
(ii) (3, – 5), (–7, 4) and (10, – 2)
Answer:i). The centroid G (x, y) of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by:
G (x, y) = ![](data:image/png;base64,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)
We have (x1, y1) = (1,3), (x2, y2) = (2, 7) and (x3, y3) = (12, – 16)
G (x, y) = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= (5, –2)
ii). The centroid G (x, y) of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by:
G (x, y) = ![](data:image/png;base64,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)
We have (x1, y1) = (3, –5), (x2, y2) = (–7, 4) and (x3, y3) = (10, –2)
G (x, y) = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= (2, –1)
Question 3.The center of a circle is at (–6, 4). If one end of a diameter of the circle is at the origin, then find the other end.
Answer:Here, one end of the diameter is at origin that is A (0,0) and let B (a, b) is the required endpoint of the diameter.
Center {O (–6, 4)} of the circle will be exactly middle of the diameter. So, to find another we have to use the mid–point formula
Midpoint of line–segment joining the points (x1, y1) and (x2, y2)
M (x, y) = M![](data:image/png;base64,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)
⇒ (–6, 4) = ![](data:image/png;base64,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)
⇒ (–6, 4) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACoAAAAiCAMAAAAeTrsEAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6Ojo6OjpmOma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kGY6kGZmkLbbkNv/tmYAtmY6tpA6tpBmtrbbttv/tv//25A627Zm27a229u22////7Zm/9uQ/9u2//+2///be6C7KAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABUUlEQVQ4T7VUaVfCMBDMQlFEjEI9aFVIhbTJ//+DJruboweiz0c+9c3Ozh6ZVIgrnu51f1Zdr09ZrF3txkwFtwSq+0OMGrmd0qyYKtQi6kasnxBhWz5wpL3D8t0LQIAwUt3UMMN6es4tULaRG1GnSp463wuFpCDbLmOnuk91ElyRMjhRiGYFcI7KchXF9WwnVJHW4hpYnOw7yRnppzCSdmJKKD7CKmksV6aghdvS67XL3thTC8YRnbqGyf0PUpRr7y9U5F88ytf+HRXbDFSgk/QZIASp1xkrs8APw+Fex1dgy+DplIu3Pw5MUBliuySFiffDpb2l/Dk+Aaz9R72htAiQ7byl0GDCPL6JJr+5HAiV04Nkz6deGIgtpd7U0I8MJLHw1WCr2WEgn5PerBoyGUi/AUeyz+6/pB1T5zZnwAV7pb4ORnorZdQAdJ8X3fwPwjefzhopaBvoVwAAAABJRU5ErkJggg==)
and ![](data:image/png;base64,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)
a = –6 × 2 and b = 4 × 2
a = –12 and b = 8
Therefore, another end point of the diameter (–12, 8)
Question 4.If the centroid of a triangle is at (1, 3) and two of its vertices are (–7, 6) and (8, 5) then find the third vertex of the triangle.
Answer:Let A (–7, 8), B (8, 5) and C (a, b) are three vertices of the triangle and centroid of the triangle is (1, 3)
The centroid G (x, y) of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by:
G (x, y) = ![](data:image/png;base64,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)
⇒ (1, 3) = ![](data:image/png;base64,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)
⇒ (1, 3) =![](data:image/png;base64,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)
and ![](data:image/png;base64,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)
1 + a = 3× 1 and 11 + b = 3 × 3
1 + a = 3 and 11 + b = 9
a = 3 – 1 and b = 9 – 11
a = 2 and b = –2
Therefore, the missing vertex of the triangle is (2, –2).
Question 5.Using the section formula, show that the points A (1,0), B (5,3), C (2,7) and D (–2, 4) are the vertices of a parallelogram taken in order.
Answer:The mid–point of diagonals AC and diagonal BD coincide.
Thus, Section Formula internally = ![](data:image/png;base64,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)
Where l = 1 and m = 1
Mid–point of diagonal AC
A (1, 0) and C (2, 7)
The mid–point of diagonal is in the ratio of 1:1
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Mid–point of diagonal AC
B (5, 3) and D (–2, 4)
The mid–point of diagonal is in the ratio of 1:1
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Two diagonals are meeting at the same point. So, the given vertex forms a parallelogram.
Question 6.Find the coordinates of the point which divides the line segment joining (3, 4) and (–6, 2) in the ratio 3: 2 externally.
Answer:Section formula externally = ![](data:image/png;base64,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)
Where l = 3 and m = 2
A (3, 4) and B (–6, 2)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= (–24, –2)
Therefore, the coordinates of point which divides the line is (–24, –2)
Question 7.Find the coordinates of the point which divides the line segment joining (–3, 5) and (4, –9) in the ratio 1: 6 internally.
Answer:Section Formula internally = ![](data:image/png;base64,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)
Where l = 1 and m = 6
A (–3, 5) and B (4, –9)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGMAAAAiCAMAAAB84ZduAAAAAXNSR0IArs4c6QAAAJNQTFRFAAAA///b//+22///tv///9u2/9uQttv/29uQkNv//7Zm27a227aQtrbb27ZmkLbbZrb/Zrbb25Bm25A6ZpDbOpDbtmY6kGaQkGZmtmYAkGY6Oma2OmaQkDqQOmZmkDo6AGa2kDoAAGaQZjpmZjoAOjpmOjo6OjoAADqQADo6ZgA6ZgAAOgA6OgAAAABmAAA6AAAAvoL35AAAAAF0Uk5TAEDm2GYAAAHUSURBVHja3VfbVoMwEMwFqqBWaKSaokIULUpp8/9f50lIIEBA6gEfmgcK7ezOzmQXKAAXtdynq/mT+jE2rlaf3hKVkw+nPkdFUFGVQQsDt6kUmXH+jC05Vhk/PRjXm311DXfqe5LXYTStUib7FsfmNRM/QBYBt4j6FKgIVVmQyWNyD9YnDwCS4tXBU6GqnEPlFAlZ0BGbapWqjLbfRwcAGtUcOpcIUNklxJDhx9DKAVmM3SKwWHXwgMtSk8PdhQCgb08n1ULULqB3B7LwEaOCyxVoDrkfRos0AML5axYZAZR/tTkAqYToDwnMcd+rLASQpfbWkfkMr27LwPRKC6B13q5XkL5hARvmcLOoCRR8MifNsd5jJJsFFXX4dXl0TBVCWATAuhzoXciENU1xm7LqWZTxU6ghuSwzWvIOQkXZPg+W5CBiYP6DQx4W5BASFubwTQ4+vpooPnEZHJey592HxhLzMX0GB29aoxxizqdH/oVDxdAcT8Oj4nxXlUt+f0AgE83XrXobDpRqA+vxqJKj/rMavmAAk4mzOQrWJhGrWX581qvUL/ZafZ4sYxTclG8T4p/TQ0Ngs/r+m80sMprXKwmK8fwyYNLepRtnfhnu3WX98fgBhLU8sKxJYYwAAAAASUVORK5CYII=)
= ![](data:image/png;base64,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)
= (–2, 3)
Therefore, the coordinates of point which divides the line is (–2, 3)
Question 8.Let A (–6, –5) and B (–6, 4) be two points such that a point P on the line AB satisfies AP =
AB. Find the point P.
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFAAAAAqCAMAAAATdiw4AAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOjqQOma2OpDbZgAAZgA6ZgBmZjoAZpCQZrbbZrb/kDoAkGYAkGY6kLaQkLbbkNv/tmYAtpA6ttv/tv/btv//25A627aQ2////7Zm/9uQ/9u2/9vb//+2///bpPYfjAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABlElEQVRIS91W21YDIQyEre1i1fXSorZo3ZvC/3+hSWC1e/EAnjx4zEMf2syQyYRQIf5ZuOZeyvWRT5WRO/Ferd7YGE0JVLXcsxESUcdNaBglY4G94lXs9A1rB51GXxjDlC0jG4zMBvg6PtH9JdwSZ/gIjaTgI2RtXwaZ09hq4TTp2b6mQ8fI4jpMQCcL2ki2ghlrVMbNGiHdiwwzah4UtdkPbc4yGSNtRUqFvTsNmpGwSzdxgiSFWFIpapKZXeEY+fG4ps3r9D5sECJslK9biDB04JNv8SzOkWjnxtvZX7WhNrL5y6oFBj/WGDMklOKMP7imDNScu0mmSG+KvUVaWusTwpjkGdJpLKujlpFBmRXOkFRhILH4Ng62LzqwYAkNyTfSGRy4Csxp4Vvo4gk8GRyOcyLkHAk+XOzisGkG/XPIuOyxE+riANORcdkjhLbCO+8/WcI/yWFncTD62hgJhVkdBay99FmIyXBPYPJz2FKx5OTfe7+O+aL+YZv99oS8Cxo/pVHD2xbPTcjo1faQkPaXUz4BxyYgH2+DmfgAAAAASUVORK5CYII=)
⇒ 9 AP = 2 AB
⇒ 9 AP = 2(AP + PB)
⇒ 9AP = 2AP + 2PB
⇒ 9AP – 2AP = 2PB
⇒ 7AP = 2 PB
⇒ ![](data:image/png;base64,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)
AP: PB = 7:2
So, P divides the line segment in the ratio is 2:7
Section Formula internally = ![](data:image/png;base64,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)
Where l = 2 and m = 7
A (–6, –5) and B (–6, 4)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= (–6, –3)
Therefore, the point P is (–6, –3).
Question 9.Find the points of trisection of the line segment joining the points A (2, –2) and B (–7, 4).
Answer:Let P and Q are the points of the intersection of the line segment joining the points A and B.
Here, AP = PQ = QB
![](data:image/jpeg;base64,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)
AP = 1 PQ = 1 QB = 1
Section Formula internally = ![](data:image/png;base64,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)
P divides line segment AB in the ratio 1:2
Where l = 1 and m = 2
A (2, –2) and B (–7, 4)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= (–1,0)
Q divides line segment AB in the ratio 2: 1
Where l = 2 and m = 1
A (2, –2) and B (–7, 4)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= (–4,2)
Therefore, the coordinates of point P (–1, 0) and Q (–4, 2)
Question 10.Find the points which divide the line segment joining A (–4 ,0) and B (0,6) into four equal parts.
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAUAAAAAzCAIAAABpM0YmAAAAAXNSR0IArs4c6QAAEYVJREFUeF7tXQlwVEd6njfzZjSXRqP7lgAJxCFAkmUMQdwIY06baAUuXLYrLM5md1OkEja72EnsLNk1FW85gQobs2BjvCa2lyOcgkUCBBISiBsJhC7QPSONRnPfVz7p2bNTo5HiHWmAR/Ur1dTTm9fdf3/dX/9HH0N5PB4OuQgCBAF2IsBlp9hEaoIAQWAAAUJg0g8IAixGgBCYxY1HRCcIEAKTPkAQYDEChMAsbjwiOkGAEJj0AYIAixEgBGZx4xHRCQKEwKQPEARYjABFFnKwuPW+n+hOl8fhGliuE8anuBT1/RI9K2+53B7I7x7sphSHEvJZV4PQIkkIHABfq92tMtgd6DscDs3lRkpomYgObTuELPdujb2uw6DS2ymKSo0Om5kWLhOzpi7gbU2LoaHLyB2krcdDjY8T5k8IFwp4IQOMZRkTAgdosNuthsPXekR8nlBAOVwcuYR+NT82UR7GsrblcBRa21dXesBemjcwEmlMTvT+V1+MkwrZQQC70/3+4Ue17cYZqZIwPtfp4jzoMr23blxuuoxtlkSo+g7xgQMg26Q013eZk6LCslPCJ8aLLtSqy+71W+yuUDVCyPI9cUP1WGXetChpe3Hm+0UTVuZGn7nTV99lgmYLWZljmbHLPTB6vj434VcbJr5flLF9fYbB6mpUWGAcjWUxbM6LEDhA66F3T0uVFM+OX54TXTQ7fkZ6eEuP2WJnWacx2lyNStPcSfLUaCFTyXlTItNjRPfajWYbO+rC5XJcTs+9Vn1prbr0nvrI9d5EOR9DKs0j/fbbfkuACEDgMB7VpbadvauueKg9d7f/ca91erpUHMYOs9NbH5PVxfFQsDy92hbdHg6w2e4a9O5ZcCFqhZGmTW2radEduNx9oFyxcW5iTno4l2WRuBBCTQgciMB83qNe88FKxe/Od31TrZyVIVs2PRrxzxC2QwiylotpsYDXo7WDriCsUmvr1TvgDydECAQ0O+oCU5/HpeZlybcsT3vvtQmIoussiEmzY/QJQZMGyJL3wQcfPJmSWFTK7ccGTLv88gcZby9IWpsfm58hgx5jkfyMqDSPMlpd1x7pIkR8+Iz/U6m81qzv7LcVz4lPkAtYUR2QtapJJ5fws1OlyVFhGEP3l3dNTZYioEiCWEwLsq9fPomeB9NtYNaR9dfKvBjMG5UNOpBdGlt1szY6nD84H8OOqoGlmMNDzJyh65oXYjMTxdVNOrj37KhA6KUkGjgAxhqjk+IijiWVsM3v9asMZk/zxsviZIJ+kwNhuTcKkgQ0xae5cREwolkwQIG3MiGdFiOMlvIxFYzaTUuRolKpUUK2eAGhpjCZBw6AMCxPq8MFH5JEO0Pd/0j+o0RgVAR2OR3K7k6D0SQMEyQlpwqEolFKQ5KPEgGDTtPTo3Q6ndHR0bHxSaPM7akkV6uUanU/zPyEhESZPOqpyDD6Qj1uF6ih1xv4fDo5JS1MJB59ngFzCJ7AZpOhvOLqsdKqR9390eHC1UtmLV88PyYmhsPC8IKiq6Oju8dksckkwsyMCRHyyBDBHdJsuzra/nCyrPx6vdlqz52UumHt0pycGVwua2a/0Onr6u7/4dSFG/Xt6Jfz87PWrVg0MXMihelgVl12q6Wi+trRs5WN7X0yiWDVgrwVyxbGxcVjNeuY1yN4H/jY8ZP//MmZCs2EDsH0h3r5pfJyGaWbmT2Fz2dHhNMLZVPjw9/s+erXn5f9/o+1Z8urXRZtRlpSeLiUXWEsXX/fP+3YvedSfwMvr5ObVV2vrL95KTcrJSEhYcw7TYgybGpo+PmHe7+5L3jMz23zjK+oedDbfGNW7lSZLCJEJYYo2wsXLvz8P4+U9aZ0CGY2W2LPX7wksPTkzZgaFjb2q3GD1MDKzkc/3f7ZWcU4QcJkPo/roej+7kfZjss7/3F91qSJgwvdPMx4g0/vDfMvg1rAb32f+6Ydeu/NxDdzv4dDcxtatF7Tt+W9jw41hnMS8mmh2GYy8FtPv//6jB9t2iiRhGMPDLNba+in9yFz493U5Xfjm9A3n6HPAz4ZITff98Ui4cEvv9hxuscQv1AgjgDEDg9lbPjjj19y/+SvNkikMpfrmQ7b8ng8jtux94tDH5/XU+MWC8NEHo7HarVTjUc/fHP6ypeX8Gi+282C1WPQXlq1Yvvub052ptMJ2Xwa+0YonaY3WXns8Md/m5c7k4Po6JheQRK4prLsp/9V0Sws4AlEFMfJoXgOp0vQfnZthj5vxjQuzfejKEMz7qAtxHx6iYd//UjufeJLctwzCb1kZlJ5c/N92fehb9FMEu8lk8nKSo79R5lGE1/IF4o5HheXFqs76/OpK//y12vi4uKcA6sGAl++vEXf8n3JS1SmzzHfet/HPfPc7+HQHHzfZF72FuSbG+5FAvrQqdJLxhxB2l9g5QNsUZ5Q2tdya6bj4usv54nEYogwpt1mjDMDgU26vqPl9xtkq7iR6agA4OFwBfaeB3mWU2uWFQiEqAILCCwQCLpaGw9eUWoz3ubyRRyPA4y1O9xCRcVv3swqenWVcKyd4SAJXH/vxk8+OnHTmSuIiOcOYE2b9JpYVemPliRMnpjhobjMkO+rwQJ2UL9O7Nsv/bSft9P7Pvd96Jf2+5QuEgmvVl66oJ3MnbhmgPNuJ0WHmbWqGMWJ1TPE8XGxnsHJYN/xxZf/3tGEGXGGjlnel4fLxG8w8v3Xd6QbKoDft1KJ6MjxkkPNMZzUBTSfPzDPS4v6Gy8vj218+7VFIDAWT44x58Y0OxDYbjF8fbrqrHoSL2EmNk6hCi4P195WsSFDsbJwHs0PYwuBVYq23x679UC0lB+RjC4FDWMymWSK0n1bCwsXzafH2sEMksAOm/nfPv7d7nKdOSKbFoW7XU674m7xZNNH7/04MSlxYK2tj/EZ8N5PNfmplP83+XDMD5jPcC+LxaJrleW/OvSgTVogjEgABwd8gZbrL8c2/2Lz2ti4WEZxDeVhQLp6ScUkGWovDM3Kl4d+yQOS1s8kYZKgdjRN11RXbN158qY5kytLpri0zaiJN9Ts2DTn9eJ1FEviWCUlZ7bsOtspyueHx4HANp0iy3Ft3y9/OOvF/DEdLkKbmcdl3/nJgR3H23QRM/kiGawhh6qxMK71kx0/S05OHvOygyQw5Hjc0vDh7oPljXYLHUnZdLnx1m3v/OWsl17i8lizXxy1cNos//rvu/dV6bXCDIoWu6yGFOud7ZsKioteowcdARZdhw4f3vVNZbs10s0VSGxdbxVm/c2mjXL2zMRYzcbf7vviy/NNfZxYjscd6+n+u/Vz1xcXCQRjH/sJabP2KDp37Pr8TK3ORMdQDuOUCN22za8WFMCOGPseFTyBAUFfT/f127XdvRq5VJSfMyV9wqSQ4hKizHsVXf+9/6vSm+1GBy+Sb3lzzdz1Ra+JJYhCs+xyO+3379fVPWy12u0T0xNzcmZKZXJ21cFhNd24dbvxcTcsi+yJqTkv5FMUa6bBfKHW9atqbt7tUKqlYkFe9qTMrGkhaohREfg7meDushJlL6ZOu0XV22eyWiLDpVgAwWGJzTl8n4DHO8bRzhD1v2GyhfzPx2r0kFNjTAj8ZBuXlEYQIAh8hwCrx2nSjASBZwUBRDuZAzTx5yeTVqtlJmVw09XVZTQaRxC6v7+/s7PTYDDgHYvFYjabRw6/Ew38rPQAIgerEWjpsSg0tgHyejzYszk+TsRsZQNpsTBr2bJlYOa5c+fwb2Rk5KpVq+Lj44fW9+7du1evXgXD5XL5K6+8gpVbN27cyM/Px8r24cAhGpjV3YYI/6wgcPxm71dVysqHmqpG7e8vKy7Va6yOgbn348eP6/V6bC/BTX19PVa2lpWVXb582Wq1+olus9k+++wz6N6kpKT79+/fuXMHNK6pqcGN3W4nBH5WWprI8Vwi0NZrHRcnLJodt2lRcoSYPndXbbS6bRbz+fPnly5dqtPpHjx4sHHjxjfeeGPdunW3bt3SaDR+OOAhdO+8efNycnKKiooyMzOxvmX+/Pnl5eXQ3oTAz2W3IZV6VhAQ8Lk4c8DmcBssrggJHS8XhIXRVdXVsbGxIpEIehUOLe4hLpjZ19c3VAM3Nzc7HI4TJ04cOHDg1KlTzLLZBQsWwK7G+8EQGE45TtYeuogWZcPPhsb3yxTlPXr0SKlUDrd0HiLCeIBZz0jf3d2NocW7Xv9ZaQoiB0Hgz0eAz6NwhumXFcrd5zpKa/txgpdUwHn48CFW1GOBtG8nH67DgzUqlWrt2rVbt27l8/lHjx6FwwxBwGTwf7hUI/nAsOZP3+7DiOJXHRgDGCdgFfg9b2tr27ZtW0lJyVBu483GxsZ9+/Z9+umnu3btKi0thUBXrlyBf88E3MhFEGA1Agg+vzAh/Aez43+4OHlpdtTVJn2nfoB7zIYAqVQqFot7e3tRx46ODgSlhm4tBNWx1jItLQ2b6vHJeM54n1kt+2drYJvT/Ulp5+eXFDjI3/cYYVgClZWVqamp2Mrjmyn87DNnzoDbENq7Z9D3hT179qSkpOzfv3/z5s1ff/014umzZs1qampqbW19xje7sbpjEeGfDAL4+TicPZaVJJ6aIlk8LdLhdLeprCmpaQa9HtRITEzMyMi4ePEiCILP7OxsBKhaWlrQ+RmW4gIdYEJXV1cjaoWvJk2aBNsbz8EOkD8gpwboPVz16tqNIgEPP6VT3ajF71l4X8P4UVtbm5eXJxR+e94/8xXEwphRWFiIUgPOXMGOnzZtGoYTfEIaGM/p6elQ4wjNjRBkezLok1IIAqNEQCTg4nebSm73nb6t+t8aVZSUTorgzpkzu7OrG7wAWZYvX45JXaguaL6FCxeCk6AMAlTQiEzRULzFxcXweI8cOQJ6Y+YJeruurg7MD2YaqeRO34rcmAVTojrUtl7dn6LYsJNhoCM+hoKhP1EAnN6GhgbYw2Dv4B7ab0cUX0SwowqeAFIxxgBIzkxnI2KuUCi8dRgliCQ5QeBpIbBwShROzH+sstxtNeJ4gvVzEtKj+DhGZ/z48VBRCPpAb23ZsgUh6HfffRemKOSEGYvZ4IHDDL67Vq5c+c4774DGeJN5B1bt4sWLB06qGuYKrIG1RseVBp1QwMWZ/jjR/3qLjpnUwqVWqzGEgI1g8sGDB3fu3IlRZO/evXC4oU4RxIJ+Zpxv3wsWP2MnM7vkoHIZHwDTYmDy0Ijc02oGUi5BIDgEls2I+odV6X+/Mv1na8b9Yu14nOaLkxWQ1VtvvQU7GQoM9xEREZgiYkxXaLKpU6fOnj3bz5KFSps+fTrDDnBNIpFgIQdjSwe8AhO4tK4fPwB36paq5La6XW3Bz23qzN/qVZQH+oGNiInDEli9ejWGFjyE0keEGZEq6GRYy7DmwWevc4sNqxACnIeyxSdywHoUCASjAl/5DkLBwUdSEQSeMgIUB4wNo7kIR/tKMm7cuBUrVsAY9hMPagwaGCzw7hsfKj/U5IYNG0Y+1SzwoXa/Pv4Yv8q37sW4+VPkWUnSq816/MJdSpQQJ09AYVZVVWF+GdofwmVlZeEG/vcLg1dPT09ubu6SJUsQW8bSE7zgO3hcu3YNz8FzmPVz584F7fEO/HskgaxPuQFI8QSB0CAAlkJLBZE39DDoMwLDkWcADdzeZ8FvAS2YHDU5SZIcJVw+Mzo9RqjU2jEnjASIPMESAFF9I1UYYGCmg8lwg7GUBMyE7gVdGcuBuWDfw49HFA4h8k2bNiETaGkoZDgJoTitLwi8SBKCAOsQCLCZoavfqjU5MxPE3l/0alSYMZM0IU7EPDl58iRmtLDaCyQcrsJwa7GaBIvIYFoP9w6icAiDgdihOGqEdS1BBCYIBIFAMLuREHa+efNmQUHBCASG7oXZAMN4uPkryIpVHLAu4LIHZ2AEUVuShCDwnCEQDIEZCGAAg3gj8HNkpGBj+5789pzBSqpDEHgyCARP4CcjHymFIEAQGAEBsh+YdA+CAIsRIARmceMR0QkC/wef3w+ZK8i+/gAAAABJRU5ErkJggg==)
Let P, Q and R are the points of the line segment joining the line segment A and B.
Here AP = PQ = QR= RB
AP = 1 PQ = 1 QR = 1 and PB = 1
Section Formula internally = ![](data:image/png;base64,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)
P divides line segment AB in the ratio 1:3
Where l = 1 and m = 3
A (–4, 0) and B (0, 6)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Section Formula internally = ![](data:image/png;base64,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)
Q divides line segment AB in the ratio 2: 2
Where l = 2 and m = 2
A (–4, 0) and B (0, 6)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= (–2, 3)
Section Formula internally = ![](data:image/png;base64,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)
R divides line segment AB in the ratio 3:1
Where l = 3 and m = 1
A (–4, 0) and B (0, 6)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Therefore, the coordinates of point P
, Q (–2, 3) and R
.
Question 11.Find the ratio in which the x–axis divides the line segment joining the points (6, 4) and (1, –7).
Answer:Let l:m be the ratio of the line segment joining the points (6,4) and (1, –7) and let p (x, 0) be the point on x–axis.
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)
Section formula internally: ![](data:image/png;base64,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)
(x, 0) = ![](data:image/png;base64,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)
(x, 0) = ![](data:image/png;base64,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)
Equating the y– coordinates
![](data:image/png;base64,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)
–7l +4 m = 0
–7l = –4m
![](data:image/png;base64,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)
l:m = 4: 7
Therefore, x–axis divides the line segment in the ratio 4: 7 internally.
Question 12.In what ratio is the line joining the points (–5, 1) and (2, 3) divided by the y–axis? Also, find the point of intersection.
Answer:![](data:image/jpeg;base64,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)
Let l:m be the ratio of the line segment joining the points (–5,1) and (2, 3) and let C (x, 0) be the point on x–axis.
Section formula internally: ![](data:image/png;base64,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)
(x, 0) = ![](data:image/png;base64,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)
(x, 0) = ![](data:image/png;base64,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)
Equating the x– coordinates
![](data:image/png;base64,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)
2l – 5m = 0
2l = 5m
![](data:image/png;base64,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)
l:m = 5: 2
Point of intersection = ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Therefore, y–axis divides the line segment in the ratio 5: 2 internally and point of intersection is
.
Question 13.Find the length of the medians of the triangle whose vertices are (1, –1), (0, 4) and (–5,3).
Answer:Let A (1, –1), B (0, 4) and C (–5, 3) are the points vertices of triangle.
Let D, E and F are the mid–points of the sides AB, BC and AC respectively.
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCADNAMIDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aiiigAooooAKKKKACiiigAooooAKwbRtV1WzOpW+oeQzu3k25jUx7QxADnG4k45IIxn2rerJk8PxuJYVvbqKzmctJaoyhTk5YA43AE5yAe5rWm0t/wDMzmm9jVHvS0gAUAAYA6ClrI0CiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAoazrNloOnPf38hSJSFAUbmdj0VR3JrN8PeNtK8R3L2lus9vcqu8RXCBS69ypBIOMjPOeaqfEbTTqPhtPLkAuILlJIEPSVjldntkMefauV+HmjXC+KUu79Pspghk8iMsGMrE7WORxgc8d8+1bxjSdJyb97ojGUqiqJJaHq1FFFYGwUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFUNa1H+zNMluFG6U/JCn95zwBQBQ/5DHif1tdK/Jpj/APEj9aybW3l/4RuLU7UE3On3UsqgfxJvO9fxH8q6XRdO/szTI7djulOXmf8AvOeSapeEwDoeDyDPL/6GaANa1uYry1iuYW3RyqGU+xqasHRSdL1O50RziLme0z/cJ5X8DW9QAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFYP/IY8T+tppX5NMf/AIkfrV/WtR/szTJbhRulPyQp/ec8AUaLp39maZHbsd0py8z/AN5zyTQBfrE8Jf8AIE/7by/+hmtusTwl/wAgT/tvL/6GaAHeI7aX7NFqdqubnT281QP40/jX8R/KtO1uYry1iuYW3RyqGU+xqUgEYPINYWik6XqdzojnEXM9pn+4Tyv4GgDeooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooqhrWo/2Zpktwo3Sn5IU/vOeAKAKH/IY8T+tppX5NMf/iR+tb1UNF07+zNMjt2O6U5eZ/7znkmr9ABWJ4S/5An/AG3l/wDQzW3WJ4S/5An/AG3l/wDQzQBt1jeI7aX7NFqdqubnT281QP40/jX8R/KtmkIBGDyDQBFa3MV5axXMLbo5VDKfY1NWDopOl6pc6I5xFzPaZ/uE8r+BreoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiqOtu8ehX7xsUdbaQqynBB2nkVV0W22N5rWN3bt5Yw812ZQ2fbef5UAbFYP/IY8T+tppX5NMf/AIkfrV/WtR/szTJbhRulPyQp/ec8AUaLp39maZFbsd0py8z/AN5zyTQBfooooAKxPCX/ACBP+28v/oZrbrE8Jf8AIE/7by/+hmgDbooooAxvEdtL9mi1O1XNzp7eaoH8afxr+I/lWna3MV5axXMLbo5VDKfY1KRkYPINYWik6XqlzojnEXM9pn+4Tyv4GgDeorF8SGQjT40SaUSXW1ooZfLZxsc4zkemevarWkweTDJ/olxalm+7PceaTx1B3NigDQooooAKKKKACiiigAooooAKKKKACiiigCO4gjureS3mXdHKhRxnGQRg1BqF/a6Ppk17dP5dtbR7mOMnA7Adz2q3WJ4wS2m8MXltcqzi4URRqhwxcn5ce4OD+FAGHpHiW18XeKYIjDNax2cTTwwz7cytnBbgkfKCOOvNdvXCaN4Av9EmTVIdVWbUI4yqxtCFiwcbgeScnA5/Sum0/XY7mf7FeRNZXy9YJD973U/xCgDVooooAKxPCX/IE/7by/8AoZrbrE8Jf8gT/tvL/wChmgDbooooAK5vxpdQ6Vp8GsliLiznXylA5l3HBQfUfyrZ1DUrTTIPOu5QgPCr1Zj6AdzXPapo2oeM7QR3jNpdojiSBAoaUuOjNngDk8e9AE2ka7pPjaJ4vJubeezdZDE77HXIIDBkPI+8OtbtpYxWQcRPM28jPmzNJ+W4nFcp4P0FPDWvX1ndT/aLqeJGhm2bFaIZyoXJ5BJJ5/lXaUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFYP8AyGPE/raaV+TTH/4kfrV/WtR/szTJbhRulPyQp/ec8AUaLp39maZFbsd0py8z/wB5zyTQBfqpqGmWmqQeTdRBwOVYcMh9Qexq3RQBz/2nUvD/AMt7v1DTx0uVGZYh/tjuPcVt211BeQLPbSrLE44ZTkVLWJc6HJbTte6JKtrOxzJCR+5m+o7H3FAG3WJ4S/5An/beX/0M1Np+uR3M/wBivImsr4dYZD973U/xCs/QtStNL8OeddyhAbiUKvVnO88AdzQB0tYt1rslxO1losQu7heHlJ/dQ/U9z7Cofs+p+IObvfp+nnpApxLKP9o/wj2ratbW3soFgtoViiXoqjAoAoafoUdvP9tvZTe3x6zSDhPZB2FatFFAGN4jtpfs0Wp2q5ubBvNUD+NP41/EfyrTtbmK8tYrmFt0cqhlPsalIyMHkGsLRT/ZeqXOiPxFzPaZ/uE/Mv4GgDeooooAKKKKACiiigAooooAKKKKACiiqGtaj/ZmmS3CjdKfkhT+854AoAof8hjxP62mlfk0x/8AiR+tb1UNF07+zNMit2O6U5eZ/wC855Jq/QAUUUUAFFVNQ1K00uDzruUIDwq9Wc+gHc1lfZ9T8Qc3e/T9PPSBTiWUf7R/hHtQBHrN1b62x02wtFvp0PM+cJbn13jv7CsPw3E+kQjUr21N9BvdftC5Z7fDEE7fQnnI5rurW1t7KBYLaFYol6KowKyvCX/IE/7by/8AoZoA1ra6gvIFntpVliccMpyKlrEudDktp2vdElFrOxy8JH7mb6jsfcVNp+ux3M/2K8iayvh1hkP3vdT/ABCgDVooooAKxvEdtL9mi1O1XNzYN5qgfxp/Gv4j+VbNIRkYPINAEVrcxXlrFcwtujlUMp9jU1YOik6XqlzojnEXM9pn+4T8y/ga3qACiiigAooooAKKKKACiiigArB/5DHif1tNK/Jpj/8AEj9av61qP9maZLcKN0p+SFP7zngCjRdO/szTIrdjulOXmf8AvOeSaAL9FFVNQ1K00uDzruUIDwq9Wc+gHc0AW6xbrXJLidrLRYRd3C8PKT+6h+p7n2FQ/Z9T8Qc3e/T9PPSBTiWUf7R/hHtW1a2tvZQLb20KxRL0VRgUAUNP0KO3n+23spvb49ZpBwnsg6KK1aKKACsTwl/yBP8AtvL/AOhmtusTwl/yBP8AtvL/AOhmgDbqpqGmWmqQeTdxBwOVYcMh9Qexq3RQBz/2nUvD/wAt7v1DTx0uFGZYh/tjuPcVt211BeQLPbSrLE44ZTkVLWJc6HJbTte6JKLWdjl4SP3M31HY+4oA26KytP12O5n+xXkTWV8OsMh4b3U/xCtWgDG8R20v2aLU7Vc3Ng3mqB/Gn8S/iP5Vp2tzFeWsVzC26OVQyn2NSkZGDyDWFop/svVLnRHOIuZ7TP8AcJ+ZfwNAG9RRRQAUUUUAFFFFABRRVDWtR/szTJbhRulPyQp/ec8AUAUP+Qx4n9bTSvyaY/8AxI/Wt6sW1a08MaJGt7MBI2WkPVpZDycDuc1D9n1PxB815v0/Tz0t1OJZR/tH+Ee1AE11rklxO1losQu7heHlJ/dQ/U9z7CpdP0KO3n+23spvb49ZpBwnsg6KKv2trb2UCwW0KxRL0VRgVNQAUUUUAFFFFABWJ4S/5An/AG3l/wDQzW3WJ4S/5An/AG3l/wDQzQBt0UUUAFFFFAFTUNMtNUg8m6iDgcqw4ZD6g9jWV9p1Lw/8t7v1DTx0uFGZYh/tjuPcV0FFAENtdQXkCz20qyxMOGU5FZniO2l+zRanarm5sG81QP40/iX8R/Km3OhyW07XuiSi1nY5eEj9zN9R2PuKm0/XIrqY2N7CbO+A5gk6P7qejCgC/a3MV5axXMLbo5VDKfY1NWDopOl6pc6I5xFzPaZ/uE/Mv4Gt6gAooooAKKKKAMzxDrMfh/Q7nU5IjN5IAWMHG9mIVRntyRzXmqePdUv9ZsW1GGxWNZCsO3cqRSNwrNkncB+HJr1HVdLtda0yfTrxS0E67W2nBHcEHsQQCPpXJaH8P9M0nX45brVWv7iFTLDbOipj+HewH3sZ46DPauim6XJLmTv0MZ+0548r06l3T73w/bz/AG291iO9vj1mkzhPZBjCitX/AISrQv8AoJQ/r/hWl9mt/wDnhH/3wKPs1v8A88I/++BXObGb/wAJVoX/AEEof1/wo/4SrQv+glD+v+FaX2a3/wCeEf8A3wKPs1v/AM8I/wDvgUAZv/CVaF/0Eof1/wAKP+Eq0L/oJQ/r/hWl9mt/+eEf/fAo+zW//PCP/vgUAZv/AAlWhf8AQSh/X/Cj/hKtC/6CUP6/4VpfZrf/AJ4R/wDfAo+zW/8Azwj/AO+BQBhar440XTdOluo7lbqVcLHBGcNIxOAMnpz37Vg+CvGtmkUmm6oqWUqlpo3DlkkBbkdAQRkfXNdjqWi6fq2ny2N3bK0Mowdo2sD1BBHIIOCDWZo3hPRPDDS3AkaSWYBDNeSKSFHO0cAD16c0AW/+Eq0L/oJQ/r/hR/wlWhf9BKH9f8K0EitZUDxxwup6Mqgg077Nb/8APCP/AL4FAGb/AMJVoX/QSh/X/Cj/AISrQv8AoJQ/r/hWl9mt/wDnhH/3wKPs1v8A88I/++BQBm/8JVoX/QSh/X/Cj/hKtC/6CUP6/wCFaX2a3/54R/8AfAo+zW//ADwj/wC+BQBm/wDCVaF/0Eof1/wqrqGr+F9Ug8m7vYHA5VgSGQ+oPY1ufZrf/nhH/wB8Cj7Nb/8APCP/AL4FAHmXiLxNNpk1jHYX0GpTxOz29y+Q0aDAZZAMbgcgDkV0XgnxpceIri4sb+3ijuoYxKrw52SJnB4OSCDjuc5rR8TeEbHxJaRRsxtJ4GLRTxKMrnggjuD6ewqPwr4NtfC/nzC4e7upwFeZ1C4UdFVR0GTnvmt06XsbW94xtU9pe/unR0VTvNW0/T3WO8vIoHcZVXbBI9asxSxzxLLDIskbjKupyCPUGseVpXsa3V7D6KKKQwrjHnvfPbxGtmDbpdbxN5nzfZgNhGzHTq/WuzpK1p1OS+hnOHN1AEEAg5B6UtFFZGgUUUUAFFFFABRRRQAVi62tu+qaOtysbR+fJkSAFf8AVN61tVDcWtvdqFubeKZQcgSIGAP40AZekC3XWtSSwCC0CxZEX3BL827GOM425xW1TIYYreMRwxJEg6KigAfgKfQAUUUUAFFFFABRRRQAUUUUAc1rNx9m8TQP/aUGn5smHmTIGDfOOBkjmr3hcn+w4xt4WSQB8YEo3n5wOwbrj3rWIB6jNLWsql4KNjNQtLmuFFFFZGh//9k=)
Mid – point formula = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGYAAAAiCAMAAACayFwqAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOma2OpDbZgAAZgA6ZgBmZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kDpmkGZmkLbbkLb/kNv/tmYAtmY6tpC2trbbttv/tv//25A625Bm27Zm27a229u22////7Zm/7aQ/9uQ/9u2//+2///bvXWOawAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACFklEQVRIS91W2XLCMAy0aaGkF+XoGXoktDSkkAb7//+ttmX5SOKSzJCHVg8wo6y10lqyTcg/tP3LupeqisnOiVteJb2wEJJdf5rIbPrUE4vgGZl60nEjC4+7sAfQPL7RwctzKVkeDZIystyEwMK6Xy5L6SApKHXyAHTdX5xo2XQxxWiXeTum86v5Jc1+KpJcYaLSA+gmP6DKSOeUnT5Y8diUKhPfPD8ihOb82WyvRft+ic6gHP1HyOZSSsbzC70a1VZ+vqR0aLuGsFmSyTQNXKO134HrMlK9H99vPB6u+ft27tOAn83XbOGKlJ7dCpyFY1Lgd+BM6kvYFPpsE415SgUl82mMnzC53lgZASnCkQb9Bs5jWYdxYwBD48RUe3jnTrSJUoVjMhaeys0p3K5U4bAan6a435W2RVYJtpkPN34Hng3EwFRpUtFeDeOqesnuTUonkEUFjn4XrmjUT6+WqanonUbphTQwjQfMVH0ICN8BrmhqLXB0ARtb4OgsoJc50o4fX0dUc1MbzwN0PG6+ncLL1GnWdVlXPBLg0dlStM43uparaB6c7YLipHsJrB6b8gmhZT9DfDioq8ZmryRvPbm/oVEt5/Hhk8Ejoa0F0EbkoNrqemxtAbQtIlBOrg/hdkQBtFtD40Mt68QSQNtnmkiWL72XExx4gqVo/R4MoEVgT4sv95qH/tPPp1aShdD7j1bL/xjoB7v9PG1evETwAAAAAElFTkSuQmCC)
Mid – point of AB = ![](data:image/png;base64,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)
D = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFQAAAAiCAMAAAD2zg4vAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOmZmOma2OpDbZgAAZgA6ZjoAZjpmZpDbZrbbZrb/kDoAkDo6kGY6kGZmkGaQkLbbkNv/tmYAtmY6trbbttv/tv//25A625Bm27Zm27a229uQ29u22////7Zm/9uQ/9u2//+2///bKCpq9wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABsUlEQVRIS9WW63KDIBCF0baxN2vSGtObSUPTNhh4/9ersOsqiJHMmJmWPzp6+Dh7EWTsn43Dy8cEjkW671Cq2/cJmIzxuy1xZLbU92o9s9DqOYqf/Iu5UsaqRDP4JXktDezzfoFQVZhF+GxfXXtD6EmZWt3oOap4QBc0k1tQ7b8VOY5taW0gByPiAhMARo01uIJTOa9d0rvjUJFidI0LyAZBZRaZsXSh9MIjfdyqIn/V+eRgFS99p+Hhc2PDFAktlk3FVHkFtcNQ6hcDhWKulOYwmelSyQwzaZYzxUOoXERx7uQS8+RKGftOIG5VaI9V0nSBd/7JD0sNF1FTp5Pn+6OI66Y5C5Rr8oSD68inhprIGyi0fH9QEEMCeg5KAz1LTs8CbT/9aapl+jSg+VVh795HVzdffcCMAAmtg1raUAYN4HETlB4MXDjd/7OIotQCrD3bSl+FHQU02KxoyPkb241+ZYOqJu7OEYjooW3UToBPRZnqp4wH7YY+VWvQtbqzUzpQIJ+q688+M3kQ06eyTnS16vwFiZopRjdun6rGWFF90U8QnMRjUK/qsAlq5T8k+gW07CelB2JJugAAAABJRU5ErkJggg==)
D = ![](data:image/png;base64,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)
Mid – point of BC = ![](data:image/png;base64,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)
E = ![](data:image/png;base64,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)
E = ![](data:image/png;base64,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)
Mid – point of AC = ![](data:image/png;base64,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)
F = ![](data:image/png;base64,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)
F = ![](data:image/png;base64,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)
F = (–2, 1)
Distance formula = √ (x1– x2)2 + (y1 – y2)2
A (1, –1) and E ![](data:image/png;base64,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)
Length of AE =![](data:image/png;base64,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)
=![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
B (0, 4) and F (–2, 1)
Length of BF =![](data:image/png;base64,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)
=![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
A (–5,3) and D![](data:image/png;base64,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)
Length of AE =![](data:image/png;base64,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)
=![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHYAAAAuCAMAAADKpB25AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmZmOma2OpDbZgAAZgA6ZgBmZjoAZmYAZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLbbkNv/tmYAtmY6trbbttuQttv/tv//25A625Bm27Zm27aQ27a229u22////7Zm/9uQ/9u2//+2///bpM0w4gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACsklEQVRYR+1XW1vbMAy1C2zZha2UsUu7ixkj7RhO4///5xZbUhLLSmK/lAfIS/iKzpF1bB0rSr08z0EBo1f6RM9YTvPtKcR12ydJ217/Kqn2343W6xIAxCaw9tN9AUt7/VMdVkUL9ewprHlbktZzNG+K06aw5sNjQbU+tL4qBEB4DLMXlPb4488Un10PazssbO2IZgZWv8ZczfsZ7epL2op6IWtEMw2j2ttN3EjuFtaD7xpFsV1WO9NyQLN/p1dfgrJTMIM7ZahqKH7/8Sb8QG+3DXHtxhvaTNpA43Z03idhaFLp8ST18W3Pck78QIN/TcDIpFixXh8sH9+47oVD3NMcv3+FPQKV+IMm1VRBONAQVGRpVZ1RLtIoZfQ5pJ2AoUkJnDxtTzlT8IjmLxSiZBialOm7l0ideRV6ld6dEss2gTReQkonw8Ck2k18jjttvNZdHnr7XUqWluwY0ewrbKApGJhUUy1XooywufG1KdJIMGVDmXauFammWrh54rQijQRDgz55WjApcUV852pBkrhauTBJSTCprLSSJBlpZ2CUdmqCDIVzvOmjadMZDcglpt2F2+7Ue4smdeq0aFI5xpfVt8KVKPUtTlJ5diG4VI5dCDCcpNyWm2Pq92JMnFYKEWFgUt1Ftei3WQYq0MRCutt1aFj0YsuMTxj+eUiqR9cRSDOCR7DD5ecQQZMUu56k4T9DELocx3AGayqva/+5RyPeUAObrvhkKRU7TIr9t0MC8xY7nIiUlQ3/6bqkxAMNwhNYu7l4dGBSYZfZoWLDf16xAw3CBZjVV+PPvXh2ZMN/3twYeiL0BsIlmNue3Y0+99xu9KHDhv/uf/JeCu3taRAuw6w+jz73HvoBnA//x9+5Wbu4h3uCT8CMXnaJgny5oU217Im5XCVx5BYlmJfYTAX+A3evTHQQsdirAAAAAElFTkSuQmCC)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADQAAAAuCAMAAABQ68okAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OgBmOjoAOjpmOjqQOmZmOma2OpDbZgAAZgA6ZgBmZjoAZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGaQkLbbkNv/tmYAtmY6ttuQtv//25A625Bm27Zm29uQ29u22////7Zm/9uQ/9u2//+2///bLHlevgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABa0lEQVRIS91V23bCIBBko7bpTRObtjG2xkrTCsL//15ZIEpS6CE5Rx/cF7OG2YXJMEvIdUcNCUSFS0O9Gk6KrEaAxHIzvJN43g0H8fsxoKf98E5shiC5vUXoTwGwOKVtNV5A8uaWprj6a17gj1h+kCbZtKmqpakVeUl46pJMM11CYzH4HbJpUwNiE3Xs2qwzYZMjyBTpgLDOoWqrapBp24IaPJJJRW7UtSIUYF44nVpBWBA1mG4n/KOjAZvI+gZJZArDVGubWiLUi4PbiBhBqP4Amd2Q3g6mLXsKOy0dGsilBeH2jnhmLv0R6w23Dv2xGCOIKIuAU82LOYRcX6tD/P1A3dvdex8QhFw/+HxXbvV1CwiClj6zbh5flev07OK4C7YIODxP0fC8ghAvO1mV7z4Tpcox/AX1xQVtov0Q+WwfFERwADHIgiPjO0WD9ISsJp/DRwaD6YiRUftP+/8N5ukYh+iMgmiLOPvCX3ZYHtYZ+2sUAAAAAElFTkSuQmCC)
= ![](data:image/png;base64,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)
Find the midpoint of the line segment joining the points
(i) (1, – 1) and (–5, 3) (ii) (0,0) and (0,4)
Answer:
(i). Midpoint of line–segment joining the points (x1, y1) and (x2, y2)
M (x, y) = M
Midpoint of line–segment joining the points (1, –1) and (–5, 3)
M (x, y) =
=
= (–2, 1)
(ii). Midpoint of line–segment joining the points (x1, y1) and (x2, y2)
M (x, y) = M
Midpoint of line–segment joining the points (0, 0) and (0, 4)
M (x, y) =
=
= (0, 2)
Question 2.
Find the centroid of the triangle whose vertices are
(i) (1,3), (2, 7) and (12, – 16)
(ii) (3, – 5), (–7, 4) and (10, – 2)
Answer:
i). The centroid G (x, y) of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by:
G (x, y) =
We have (x1, y1) = (1,3), (x2, y2) = (2, 7) and (x3, y3) = (12, – 16)
G (x, y) =
=
= (5, –2)
ii). The centroid G (x, y) of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by:
G (x, y) =
We have (x1, y1) = (3, –5), (x2, y2) = (–7, 4) and (x3, y3) = (10, –2)
G (x, y) =
=
= (2, –1)
Question 3.
The center of a circle is at (–6, 4). If one end of a diameter of the circle is at the origin, then find the other end.
Answer:
Here, one end of the diameter is at origin that is A (0,0) and let B (a, b) is the required endpoint of the diameter.
Center {O (–6, 4)} of the circle will be exactly middle of the diameter. So, to find another we have to use the mid–point formula
Midpoint of line–segment joining the points (x1, y1) and (x2, y2)
M (x, y) = M
⇒ (–6, 4) =
⇒ (–6, 4) =
and
a = –6 × 2 and b = 4 × 2
a = –12 and b = 8
Therefore, another end point of the diameter (–12, 8)
Question 4.
If the centroid of a triangle is at (1, 3) and two of its vertices are (–7, 6) and (8, 5) then find the third vertex of the triangle.
Answer:
Let A (–7, 8), B (8, 5) and C (a, b) are three vertices of the triangle and centroid of the triangle is (1, 3)
The centroid G (x, y) of a triangle whose vertices are (x1, y1), (x2, y2) and (x3, y3) is given by:
G (x, y) =
⇒ (1, 3) =
⇒ (1, 3) =
and
1 + a = 3× 1 and 11 + b = 3 × 3
1 + a = 3 and 11 + b = 9
a = 3 – 1 and b = 9 – 11
a = 2 and b = –2
Therefore, the missing vertex of the triangle is (2, –2).
Question 5.
Using the section formula, show that the points A (1,0), B (5,3), C (2,7) and D (–2, 4) are the vertices of a parallelogram taken in order.
Answer:
The mid–point of diagonals AC and diagonal BD coincide.
Thus, Section Formula internally =
Where l = 1 and m = 1
Mid–point of diagonal AC
A (1, 0) and C (2, 7)
The mid–point of diagonal is in the ratio of 1:1
=
=
=
Mid–point of diagonal AC
B (5, 3) and D (–2, 4)
The mid–point of diagonal is in the ratio of 1:1
=
=
=
Two diagonals are meeting at the same point. So, the given vertex forms a parallelogram.
Question 6.
Find the coordinates of the point which divides the line segment joining (3, 4) and (–6, 2) in the ratio 3: 2 externally.
Answer:
Section formula externally =
Where l = 3 and m = 2
A (3, 4) and B (–6, 2)
=
=
= (–24, –2)
Therefore, the coordinates of point which divides the line is (–24, –2)
Question 7.
Find the coordinates of the point which divides the line segment joining (–3, 5) and (4, –9) in the ratio 1: 6 internally.
Answer:
Section Formula internally =
Where l = 1 and m = 6
A (–3, 5) and B (4, –9)
=
=
=
= (–2, 3)
Therefore, the coordinates of point which divides the line is (–2, 3)
Question 8.
Let A (–6, –5) and B (–6, 4) be two points such that a point P on the line AB satisfies AP = AB. Find the point P.
Answer:
⇒ 9 AP = 2 AB
⇒ 9 AP = 2(AP + PB)
⇒ 9AP = 2AP + 2PB
⇒ 9AP – 2AP = 2PB
⇒ 7AP = 2 PB
⇒
AP: PB = 7:2
So, P divides the line segment in the ratio is 2:7
Section Formula internally =
Where l = 2 and m = 7
A (–6, –5) and B (–6, 4)
=
=
=
= (–6, –3)
Therefore, the point P is (–6, –3).
Question 9.
Find the points of trisection of the line segment joining the points A (2, –2) and B (–7, 4).
Answer:
Let P and Q are the points of the intersection of the line segment joining the points A and B.
Here, AP = PQ = QB
AP = 1 PQ = 1 QB = 1
Section Formula internally =
P divides line segment AB in the ratio 1:2
Where l = 1 and m = 2
A (2, –2) and B (–7, 4)
=
=
=
= (–1,0)
Q divides line segment AB in the ratio 2: 1
Where l = 2 and m = 1
A (2, –2) and B (–7, 4)
=
=
=
= (–4,2)
Therefore, the coordinates of point P (–1, 0) and Q (–4, 2)
Question 10.
Find the points which divide the line segment joining A (–4 ,0) and B (0,6) into four equal parts.
Answer:
Let P, Q and R are the points of the line segment joining the line segment A and B.
Here AP = PQ = QR= RB
AP = 1 PQ = 1 QR = 1 and PB = 1
Section Formula internally =
P divides line segment AB in the ratio 1:3
Where l = 1 and m = 3
A (–4, 0) and B (0, 6)
=
=
=
=
Section Formula internally =
Q divides line segment AB in the ratio 2: 2
Where l = 2 and m = 2
A (–4, 0) and B (0, 6)
=
=
=
= (–2, 3)
Section Formula internally =
R divides line segment AB in the ratio 3:1
Where l = 3 and m = 1
A (–4, 0) and B (0, 6)
=
=
=
=
Therefore, the coordinates of point P, Q (–2, 3) and R
.
Question 11.
Find the ratio in which the x–axis divides the line segment joining the points (6, 4) and (1, –7).
Answer:
Let l:m be the ratio of the line segment joining the points (6,4) and (1, –7) and let p (x, 0) be the point on x–axis.
Section formula internally:
(x, 0) =
(x, 0) =
Equating the y– coordinates
–7l +4 m = 0
–7l = –4m
l:m = 4: 7
Therefore, x–axis divides the line segment in the ratio 4: 7 internally.
Question 12.
In what ratio is the line joining the points (–5, 1) and (2, 3) divided by the y–axis? Also, find the point of intersection.
Answer:
Let l:m be the ratio of the line segment joining the points (–5,1) and (2, 3) and let C (x, 0) be the point on x–axis.
Section formula internally:
(x, 0) =
(x, 0) =
Equating the x– coordinates
2l – 5m = 0
2l = 5m
l:m = 5: 2
Point of intersection =
=
=
Therefore, y–axis divides the line segment in the ratio 5: 2 internally and point of intersection is .
Question 13.
Find the length of the medians of the triangle whose vertices are (1, –1), (0, 4) and (–5,3).
Answer:
Let A (1, –1), B (0, 4) and C (–5, 3) are the points vertices of triangle.
Let D, E and F are the mid–points of the sides AB, BC and AC respectively.
Mid – point formula =
Mid – point of AB =
D =
D =
Mid – point of BC =
E =
E =
Mid – point of AC =
F =
F =
F = (–2, 1)
Distance formula = √ (x1– x2)2 + (y1 – y2)2
A (1, –1) and E
Length of AE =
=
=
=
=
=
B (0, 4) and F (–2, 1)
Length of BF =
=
=
=
A (–5,3) and D
Length of AE =
=
=
=
=
=
Exercise 5.2
Question 1.Find the area of the triangle formed by the points
(0, 0), (3, 0) and (0, 2)
Answer:(0, 0), (3, 0) and (0, 2)
Area of triangle = ![](data:image/png;base64,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)
x1 = 0, x2 = 3 and x3 = 0
y1 = 0, y2 = 0 and y3 = 2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 3sq. units
Question 2.Find the area of the triangle formed by the points
(5, 2), (3, –5) and (–5, –1)
Answer:(5, 2), (3, –5) and (–5, –1)
Let A (–5, –1) B (3, –5) and C (5, 2) be the vertices of triangle.
Area of triangle = ![](data:image/png;base64,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)
x1 = –5, x2 = 3 and x3 = 5
y1 = –1, y2 = –5 and y3 = 2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 32 sq. units
Question 3.Find the area of the triangle formed by the points
(–4, –5), (4, 5) and (–1, –6)
Answer:(–4, –5), (4, 5) and (–1, –6)
Let A (–4, –5) B (–1, –6) and C (4,5) be the vertices of triangle.
Area of triangle = ![](data:image/png;base64,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)
x1 = –4, x2 = –1 and x3 = 4
y1 = –5, y2 = –6 and y3 = 5
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAmwAAAAqCAMAAAA51fdPAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6Ojo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmZmZmaQZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bUprpWAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFk0lEQVR4Xu1ba3fbNgyVmiWLuqZZvKRbO61dtabuslWy9f9/XCk+JEACX6aZqkfwl5xYNghcXAIgCBcFvxgBRoAR2AgCTw8fN2Ipm/mdETjclxefv7MOvPw2EOheP+6ZbNvw9RqsZLKtwQsb0YHJthFHr8FMJtsavLARHZhsG3H0Gsxksq3BCxvRgcm2EUevwUySbIeHd2vQbRU6tK8ePXowWiNAHrD2L5Y3CO1LvlWYCNZV7p3HaIHN6ACrf1+VZfnTW7x1O+YaAqSr/nTENkYrAqwFjn19rd/rP92c5S5r/9qTiU5eKJ/kAojeX321su3saD2PSbQ95/CDCyy46tN9eVcUrUmsff2rHeSoeqr7xRUbihMX+iKSfR7JBRbtCm3nRSujSYPD8qAVDhYgTVtKcjV6H/e1YB7xoqo8G/X6fyopzZlrLAs55lG6IfuXclOESg7VmhI9Ra+lnR60godqMpoklE7xQ2GHLhasCb6+lrw47nQWbekr+q4ijhQWsrXV7f/qEQqufY0CHbmQcx4FhhoctrHoSXKw1qRoe3/IjVbEUE1Gk0SyOt0PYjvbHR4JFmCJxq2rVECzbOfj7veRbMc3qqQ76L9zyrXTOQ4FIMwIciE0j7JYBxqJQxsSPUkO15oUPebKxaZyooWHatxopZiU0Q9D8LE7PBIsgJ52TVuqsGMpVJq7Cfm+vhzYdtzdkpFt3PWSuiAnY7LZKqIpnCzWgV/BSRiJnj4WrjUp2l60edAqQEx0o5ViUlY/uKCLBAuyRJUfhkv0bm5vvoIHMvNCTiHSGUfINxtzxJXMg2nUFjaQn/A6yPlQMhY9So7QmhRt4tdyT7nREuUDqEWcaCWZlNEPTugcYDVDVS1fdNWlgDHlIMmB428fp9OqpM3Vf/fXljNrc/Hhvixv1VNTSKsMHUm2+Tpd9aoqL3XnGUqmyRajNSnaTjYnWgOasPB1oZVmUjY/uKGLAWskXwmKNAOfgQlSVKYswcL9SJa+LkETCtFZPBKn2yddAhpKLBlPLTToM/MTXKf/611xqPWWGcm2EK0leLTGXyNFG7IRymslgowQW8GOVppJmfzggc4JFlla6Te76mcZg1yJoTX0tJANydenW82FtMg2c5MuKlUZ6I1sUVqTos+TRoeQDsm29MZIathFRnlAe2dhkkNyih9CoJsrrf73pNHZAcF6AoMPZPi29Ne1kXrPJ9VsKo3O1jG1YkDNBvaQiuF2rSUD5qL9B4SAwvOEdaUyFNnkA1w8Z/PDsm7CjggHC+4uRQtDVCvAxAGBLtoaKU8FnuMu4TQ6sEMeRMw6zSBMK4okY89AE0K1JkX7Wx/+I7XkGrICRbYkk5ySi0Q/zEgNTYgECxpsaKHOjda2+dRQnreBZ3mhqwQ7/lWdlNg+m/iKY51GVEqHWtXeQX02pzTsdEq0v6lrQwt0391opZiU1w8OR5BKB01IGrxMDWS5QRBp3BxmD3+oUNPpvzOyFZ04jF7+Ld+NvUFA8yiLdQ4PonjUt/thNwjBWlOiQ66rKLTwUI0brRSTMvphyNZ2h0eB1Yur97GFYHKvyRm2u9E5pUL+D73BDJGFP5NP8hQzQy7iz4jWc5hEAZ1kwqi0FaymFC2EnW4F6WA85c8ThzEIO/JMG8igmmWeRMXrUXTQiNHZ0HoWk8hdnWBCAFjyILfX91MiXqqRopGkJ483zUzJN6KVTzKYZwscnjwTWs9ikiWDnGzCpLQbLMEx0zVT82wiS/OoLvAGj4VHFDc+sNTJGL34JxwTHPyDlwiu+cDyxL2IlfijjIAbgaQjCIPLCEQg4GohRYjhjzICfgQa23yQ/6v8CUYgCgHZQWrpH7ZECeIPMwIeBGRHrZfXqfxiBPIioIeOmGx5YWbpjAAjwAgwAj8+At8Apf35zr8qRn8AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 19 sq. units
Question 4.Vertices of the triangles taken in order and their areas are given below. In each of the following find the value of a.
Vertices: (0, 0), (4, a), (6, 4)
Area (in sq. units): 17
Answer:Vertices of triangle A (0, 0), B (4, a) and C (6, 4)
Area of triangle = 17 sq. units
Area of triangle = ![](data:image/png;base64,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)
x1 = 0, x2 = 4 and x3 = 6
y1 = 0, y2 = a and y3 = 4
⇒ ![](data:image/png;base64,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)
⇒![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 17 × 2 = 16 – 6a
⇒ 34 = 16 – 6a
⇒ 34 + 6a = 16
⇒ 6a = 16 – 34
⇒ 6a = – 18
⇒ ![](data:image/png;base64,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)
⇒ a = –3
Therefore, the required vertices are (4, –3)
Question 5.Vertices of the triangles taken in order and their areas are given below. In each of the following find the value of a.
Vertices: (a, a), (4, 5), (6, –1)
Area (in sq. units): 9
Answer:Vertices of triangle A (a, a), B (4, 5) and C (6, –1)
Area of triangle = 9 sq. units
Area of triangle = ![](data:image/png;base64,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)
x1 = a, x2 = 4 and x3 = 6
y1 = a, y2 = 5 and y3 = –1
⇒ ![](data:image/png;base64,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)
⇒![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 9 × 2 = 8a – 34
⇒ 18 = 8a – 34
⇒ 8a = 18 +34
⇒ 8a = 52
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Therefore, the required vertices are ![](data:image/png;base64,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)
Question 6.Vertices of the triangles taken in order and their areas are given below. In each of the following find the value of a.
Vertices: (a, –3), (3, a), (–1,5)
Area (in sq. units): 12
Answer:Vertices of triangle A (a, –3), B (3, a) and C (–1, 5)
Area of triangle = 12 sq. units
Area of triangle = ![](data:image/png;base64,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)
x1 = a, x2 = 3 and x3 = –1
y1 = –3, y2 = a and y3 = 5
⇒ ![](data:image/png;base64,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)
⇒![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 12 × 2 = a2 – 4a + 27
⇒24 = a2 – 4a + 27
⇒ a2 – 4a + 27 – 24 = 0
⇒ a2 – 4a + 3
⇒ a2 – 3a – a + 3 = 0
⇒ a (a – 3) – (a – 3) = 0
⇒ (a – 3) (a – 1) = 0
a – 3 = 0 or a – 1 = 0
a = 3 or a = 1
Therefore, the required vertices are (3, –3) or (1, –3)
Question 7.Determine if the following set of points are collinear or not.
(4, 3), (1, 2) and (–2, 1)
Answer:(4, 3), (1, 2) and (–2, 1)
Let A (4, 3) B (1, 2) and C (–2, 1) be the vertices of triangle.
Area of triangle = ![](data:image/png;base64,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)
x1 = 4, x2 = 1 and x3 = –2
y1 = 3, y2 = 2 and y3 = 1
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAhEAAAAgCAMAAACb+doBAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmZmZmaQZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///b1w+/fgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFf0lEQVR4Xu1a4XrTOgx1BgMKXBhQOu6FWyAwwgZpmvd/ORLbkmVLdhUo+/at7p+ty3J8cnQsyXaMqZ+qQFWgKlBUYPzypCpUFQgKfH+5qY6ohogU6NARw2XzBi8Nlx/uu1D9P1cHHvEERDCgQteAHOiI/frJT1Sof/r5vhvCmN2q7PqTEAFVGNsH32zM0RGd/8P8x90pGGK2xP8F35+ICKjCfu1KBDhi3IaGIvw+fn3hjKP5dBcH0vAStKOCxXUy0OwehbSYPiER5M7xlqOxhKYh+oIKrVVjbB87Uagj+jNfM8bt67xknNbueWnOTUMsQTsqWMQ1IJeSBIpwB3mLjlgmr+EqWEd0zfSxyWJqI3AcZ5bZJdhqdmASTma82TTNufVQMdEGtJvLUpcCV5VggRAhIkg2Xj9jNOk0SG8BEYgKBd4SuhQ3IilePsB7sbymEKxpUKYvqACNhCdGxEFz9Nha7FZ5R7TNBzOs3b9GaXjcRikD0IZNQ1qWVDZyVQNGb6dEeDTas09m2LrHQGTaPMW3hBmi4i2iu1klirCAd07eBFoXLCPp61XAOeDJhe+7lUsNwST79b/oiP0711kM/qdp5+TSNfbBo3kdiwFou4urEAcGRq8qwKIoUiJT0kt4ijRDaUgtBCKgCmXeC0X4U94+PsRsymCJ+joVWMIMYepddEkr3r6hVfV8tsR+/Yo+lr8nSoqxI0jNDkONWw6GV3VgSSiB/PSAAk+fGBE530ggjpq3MSk6zxGK8UTekrxxjgjIh4LF9XX3wloDBQ05GqKPLuhf/CSTadxObQZtO2YIqEF2rvhP7AiCQHK1ABauqsASR4RiyKGHj7DMAGTMBGmKMOHhQ70s8+bo3BH5nFTiLcsbO0IfLK6vUyFtcQa/Gp0rgNcABtm//RwEsrc++rEh21k0ndBapHGEAEYYk8VhFiyOJZ2ECc/9umku/NIJaOYdgd2Z7GTGW0Jf4IgC75y8siMOB4vr61XoHlyZeaUxfeyYYXKDGP5W651Jlw6r1rht4oV8sBdI3XropoEpRiZY1M8xMM44D5ZeiX3OoG9WvqdljmBDoCPUvA1DPxZvLq/hfN2TKYKVdYRpz+j0Gja40kyqRg/BzTqC9CSLc8T0DIm9/iBHJM0Rgza9P7o5nCMOVI1Z+4T3NG0SdHWOKPPOyCvmiMPBMvkccR7vMPLOkhY9+rvNxjRJtKGEqEo/zREMjDLWNCXU1oQIVLc4mUGVCH1Ebk8NO8tS1Ui2PFN0tSPKvDPyZvqIadBisAR9M50lX33S2iZ0lribaXvS3u9zhePTZL0s9uxzObZtKtkaRb9EzW924UIcQYg4QxBol3l9nBFZsfpU8ZbRtWuNEm+/eSLIm1trRI5Q6SuvPoUdKprLwjaYHYPucdptg7G1VtBtIRTASGOrBAuOoES8IQLPcd6cGhz3gKzYoSIq5HnL6NwR4iZpkbfjKskbO0IZLFFfp4IPLOpJESFdhG2wqTxBhzj85+bzzv+EJsc6QrHNOH5cTY3Jw/cWhIHRqwowWjIiIhx6mMd95bgDsmoX26tQ5i2h26FUe5a2U/PZNZXE95Bc3uyeZSlYkr64ix0XQZqiIZdKm/BRDNIvi48iSmhHBYsGQmTVSddSFW6BtyTbUpohUYIKdkvkx3RI5fYexdPw3z5OEyO9DO02zj51p+F3j/cR5CVnn/Q0fP/2k7n25YCuG9Hmv3vkLs/9JWi38X6E9o2Zu8Y7k1mX0CTvR4AKmBMg+ruVL7LzeCfxQll9q872g/7dwvEL7iXCerG+eZvOvdN88/Y6OsIs9o314iko0FVDnEKY9c/YT4boiy9H6rHqf94DBeYj3Ma/H3MPHqc+QlWgKlAVqAr8TQV+AUlY+IuM3e49AAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN0AAAAgCAMAAAChFo3vAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGYAkGY6kLaQkLbbkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29u229v/2////7Zm/9uQ/9u2//+2///bSpEENAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC00lEQVRYR+1Ya1fbMAyNy9jINlagdK+MtWFjHt1wGv//HzfbkhMpjdVmJxxoqb+EVLaurp4OWXZce+wBe3u2x9ZvMf33h/kBs8syfWS3t8n7UmK3mqsrHyT78/zXaMHS0ztR16OAaXW6CKi2fPMA8EZdhr9sAc+RVvX+q6DpkcDsD+VRtXILIla8RnLhNbt/p9Tks2DY6uNSjsptHjRWb9P7bAHYHuxUVKcnMhrKI1hkEy2sZ9A8zUlISz354hgCcVi2oEFYzxVsTC2TX/wFmQa39S0EKyeLbF1I9lc5k3JbnOZGHsFKbp0tAjv+KIlhTGM1vdMiO5M778BKBw/BstJD65BM/auefRLZtfIIRi33OuG9ygMG4gZYXF1/iexiJoAukgHMegSD3zreZhvLKyOya+Ux1yFW7QJroxajfGZSlYPYGRoI6iMGSUxe3wjN1Zw/iOyoHMDqGfcoRqvRsspdV1kSYwaxK0++z5W6gILrZkmjtAGrZ0pNk9VZXy8br8NZbguTI5h/+I7plttf5TAZYr5VuSPflEKJGylhkpkbYlv48bLyOgi7jW00uVd5rOPutpBszhEa6rIHjMqRnf02oWnCYwcdlfXVIbHDg8SPvArwjaWboQ2abjfRtW3XYbZweQKT1R12BVrp/8EOY7NL3fnsSTUfT1XuKkTeX3fdnhk6aHIikBzujQq4BY53K7w90DRoz4s1og2lu7JDsDhroh4+6FzJucIRprmf98teYuHHKj/zx0MubZ131g/yNd6VEjo52sY0b6xJzLvOXSXcxPAuCoBMo71xLVW9St/UKtcy8fj2u8raa8MG28/O1ZY0nVzgUZ68q/B7ZjouAyU73DMHahS283vmHz+VYPfhfSPU14v2QnKA33fyF8p4KfNUmrQ0Zp7KqLFw77HsxtL3rPToQyZnHDkj/ePjWYVioDH+y0MJ38QD1R23Hz0wjgf+ASXTVa4tLK3IAAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Therefore, the given points are collinear.
Question 8.Determine if the following set of points are collinear or not.
(–2, –2), (–6, –2) and (–2, 2)
Answer:(–2, –2), (–6, –2) and (–2, 2)
Let A (–2, –2) B (–6, –2) and C (–2, 2) be the vertices of triangle.
Area of triangle = ![](data:image/png;base64,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)
x1 = –2, x2 = –6 and x3 = –2
y1 = –2, y2 = –2 and y3 = 2
⇒![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFkAAAAgCAMAAABO+CSUAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGYAkGY6kLbbkNv/tmYAtmY6tpA6ttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bFGeiNgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABgUlEQVRIS+2U61LDIBCFQa02XpvGW7SlKraWFN7/9dxdyIUEMk1txnHG/ZUA+fZwdrOM/ckwq+k4uj+vs5HIjMl/cqNoo7lhxMXW5dlkfPbjPpF8siCI5BCWp/hdmcPjy5NlT77NPe0akMUn+Gje+LN/3uTnQXCR9JB3GT/9QI7gT2yX0nMHpNNg++n0IU4ubt6lI+PH0qoVdqkKkwfJYqZ63XBk56clty/fWSDzr7b7k63YrsRm9vImer5ke5OLhCTrFDsCOwOCqmvdEG6JczDB5HAIyNJW29tzyWs9dDjgRpFUfV17r8o0rT5qlKci1yaYF78ykQqSZq/U/ktFFtOyaTsFC/lMRdyLLLEhFPphfW5GsDfgQP8/6HaLS7iYEcjsXD7mBngdFW1eE6jE2WNVXFLblhj5B3ssjm3Ztv6CQXJLR2JzYzDagvR8wdbusrFZNxDdmHVkPsZx5zPWvt0kA0VGj6+tzccPORZYAVjFp8LhN9EpzpwxyIdr+rUvvwGQnx39T/y3tAAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
Therefore, the given points are non – collinear.
Question 9.Determine if the following set of points are collinear or not.
, (6, –2) and (–3, 4)
Answer:![](data:image/png;base64,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)
Let A
B (6, –2) and C (3, –4) be the vertices of triangle.
Area of triangle = ![](data:image/png;base64,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)
, x2 = 6 and x3 = 3
y1 = 3, y2 = –2 and y3 = 4
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAjcAAAAoCAMAAAA16Sv7AAAAAXNSR0IArs4c6QAAAKhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZmZmZmaQZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLaQkLbbkNv/tmYAtmY6tpBmtrbbttv/tv//25A625Bm27Zm27aQ27a229u229v/2////7Zm/9uQ/9u2//+2///bo5IcvAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAHu0lEQVR4Xu1c63rbNgyV0nWr18ua1c1unZstVbpNSdZVlvX+bzZJJHEhQQqSYjv2J//JhRQIHhyCAEg5y5bPgsCCwILAE0Cg+ePvJ6DFQVXYXg4Pt8AygNHtu2EQz62HYs6KLueGSjZmzuWLs5v+8ISaze8DnRZYQoDqn5794/67/f6T+7V+nA2ruvw6bLhj96gQAVEVAsuxVdWOP9l8xGAWlvJCsmG1+hEtW4C72b4CBmlVlfuVr4GV8wTt8elmk96dEZY9KvGoomeYDw3mYKmvvrUUaW4dQXZrsjPhutqtJdd99zK/+G3s/Eo36NgHD9g/7XBOz93MMh8azMGyW5t1dffDlWNLdUH8Cj4hrrDm4012T/urLDu0mFVC9tzJARNxmSfAfK75LPOhwQCWwiEAgV5JtnZEL77CJqy9oehhz5xQiQdghN5pUqnEH7jTTPOhwRws4F2ANxQw9D3RDb2+/jAag1NwOMztejNMtY0G4yAPzDQfGsxNvcpt2CLyBnzPdhVJTIv8m/G8yahPOwhu4wfZruKR8Qmozyc823yECAYWkIi8wX2q2cAuFs9LH2KUStgqOo/+mebzG3XGVb4dqGZPFdZsooUrhOWIunJ0B2BIEF1nPjCYg8X5m6b4zmZWZAxMrSK7/e79J2TeiDWdDBCaDakDDArdvkqW6KYLiwc4JOOcLl6Y2DhhnkdJwjDbfGgwK8ruV2XefowHKjE/Al/NcnOq8N1Km4c3D1d5/tymanTF+gDiXtrcrlJ5i1U0GZdbYc39Sxw8sBhpJcKKqIvFLUylqyw+5A2J+ogNkguHgJqCYbL5QA80mIUl1BAiniyDX1Ob/aBHMB2K/ENWr50tmFF4VR9i92p1+V9yr7ME5+UgUVhxcZPVm1jBgLaisLjtECGVrrL4fkNmxxmYtGxXytoGBZXBwCVPNR/RAwxmYQlXFYk9ACDCpagldz+boKS2P72OfURfQhROgWGThLiiWmHALcjerX+xMvhKE4XxwX1ptBWFKXgj6prpxAe8wYAKp5YFwjiqsuYhIx3sqTUeYkz1ADQcb4KdgJgBOmt402yed8TZrROXEDB7i/LG0Zb5VkF28c6lhDyrZ7xh8TcsEVFT20oyzijYAIuoa+tFBCAC8QFvUFecWueSBlG1oDIYPE+m4U04EtXDp0IYMRF/55MsvSX1e2B0KzWbldunwPEE8AUFgv7BQHb15itUUVhxgiFGCi31NWZeoabYCsLiawUxtOz3uqrERyee0akJMw+M4EClMHDe6MIlX22mBxjMztXP0JrPr/GYYRxvOuN+uXqROO3GJcWQFk1dPPuzDaTh3NWT3SVyyBvqM2Xe7NZ5/hY186WRVlhII3jj6eoDQQdn61TUlU3NLJkkqgAqFT2FN95IXA8/ZDEJVpdNtZ9OyzRvCtszz/ulhs9Zr5ATE3p9OwSglAZqBJ0skZtN3mbjD1h8a/+BsntJLW/KPgcHwGLCeuUeViRBYtJYa8ibUKrbZGO6tjOlQNDBB3XlUzNeKYUqgupEB/piipw2HxvJ0yMIdeu1F+FI+5Qmvglm6DtUWkljnlNadjbxwzXE0KscfTlv4r6/a6lsoSG0BW2d4G8CXUPewOCD/oZPTdaVIEtAjfsbpfmY2p4efnwTVuBILDkqLrYeNV5xKcgWpuUNbKO9t+ayx8U3XWkc3J0gzbVOiG8sb3DL14hPcpwegAnCCG8IqPH4RsmbcCTUw6dCeBdGysPT5wJ2Gi6uigQ4/UiVNV28fuOGMj3cGpJku0lx6kv5lPHliB6XxlpRmCIPl3WFIN4AEREf8IZiTHiTRpWAymDw6zdD116NNiavIfYj59qQ0ZhNOsynptb97KqLJVR9et8UjjfRUNY53u2qncC9TSBF2c6ww/Wbpiv51XDY5kljrZPqN0xXg39XkDAHXBHxAW/kbTyNKgWVwTCh7ieNhGsH79vEeCOdMyQO+cBn1r8aom7tTy+8sSGZ4Q2XJ5dNt925xI2RIsluN2AzBUW9uL5ekdzMl0ZbyUU1xTmDK/FSXQNlZfEBbzKsF8PU5JkjshTURL1YYz4JY9QDJbhzhgCdMeeaHjfUf6YK32Mv52jOp/SK4TV8zbnmUXXlk0rCkLqEpsIGDWZFWe/ypauVGAkk9MDTrNhNJfKcanzbKXnxadyx8L7Ow3X3KI6oq0eb5Hn4bPOBALhH0fv63Xu8J0y5Cb4ncvGBPjeGN2n+T70yI2owVZjy3tZU8bN15QIG7t/MNh8YzMGCUbxzdPI90fg7CBPuF8uX68cQb/99z+ue6EzzocEcLMib0qY68r30uKndcyNMeQovwpzXvfSZ5iPZgk2EYVndu2PsyHswMVvDc3renIK7Obf3YGaZDw0GsDjvUsLth8h7d5l8JR6fU/NmbBKiFvyYHc/tvbs55qPnijb9diSpWtpUtqhYdSeK7oNsaT4KLwXT55Rma+Uoex6x2xC3T+893+nmIwaD93w3hj/dSX8Ol3u28vcKZP8Gbxnw53R2rv/S9Ttqr6E3AyckA0edTzf4VPMRg8H3Cgy9RdJWY5fvMZEsvsAytA7GfFnOkKxTaVfMWdHlVGar1nPUnJcvJJNrdI/zbUBqmz2Bjpqvr3sCai4qLAgsCJwjAv8DXD1BKi19phoAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Therefore, the given points are collinear
Question 10.In each of the following, find the value of k for which the given points are collinear.
(k, –1), (2, 1) and (4, 5)
Answer:(k, –1), (2, 1) and (4, 5)
Area of triangle = ![](data:image/png;base64,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)
x1 = k, x2 = 2 and x3 = 4
y1 = –1, y2 = 1 and y3 = 5
if points are collinear then, area of triangles are collinear.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ k + 6 – 2 – 5k = 0
⇒4 – 4 k = 0
⇒ 4 = 4k
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAAgCAMAAABXc8oyAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOjqQOma2OpDbZjpmZrbbZrb/kDoAkDo6kGaQkNv/tmYAttv/tv//25A627Zm29uQ2////7Zm/9uQ//+2///bhIvt3AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAoklEQVQ4T92TzRKCMAyEWxTqD1gEqVLI+7+maStwgDF7YMYZc/7Y3WyJUj+ZwdwgX6pPGOgqC4H+QhA4lh3Z6i6HdDpMrpzOGolO1l4GX+bQQWByBBQXkAMfH1LSoEi26CUuWvvzxLWxjDCrHRl8ztyG7Oc7XibUKRvHrQdTTFLfrVkV+D0i1MrPwxGzhqzWV7GfHYH/uxmkHOxcWSndzN7zBto8CLgLRf6uAAAAAElFTkSuQmCC)
⇒ k = 1
Question 11.In each of the following, find the value of k for which the given points are collinear.
(2, – 5), (3, – 4) and (9, k)
Answer:(2, –5), (3, 4) and (9, k)
Area of triangle = ![](data:image/png;base64,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)
x1 = 2, x2 = 3 and x3 = 9
y1 = –5, y2 = –4 and y3 = k
if points are collinear then, area of triangles are collinear.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAPgAAAAgCAMAAAAIYkV7AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bpw2KSAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADaUlEQVRoQ+1Y63bTMAyOBwXCpXSsdECgJcDIWpakfv+Xw5JlW059Y2XnbKf1j61zLVnfp6tXVed1ZmDKgPzx6iRJuX2/Ok3gVdWdgZ9YyJ897jtc/pr/PiYEusXNMeIPLyvbl3fmlt1KXNFn2Xyw22qrExffs7bcekeGt19SEjFifSWZO61ZhuZOzDZaRq6FmCe474RahLYXFqxscE9u3wgxAzx9EvhQgxo44iSq4XWCqwmxcJlTov7YXaeElYcmZhma5U+BfMvmxd3YPCsKWjhrGO61SHuxqcYGAOWAG+86CRUmTt/Uc0Sstz3UNkTGlfBtlg0Pn1Z8rsYlnHBmGZoJBX7RaxJya7+0jU02+mMLvzoQLwZuJZQL4y4nYiPAh8VNlwQeMsvQ3KIk/mSIUuANWow6RhXqAOAqLWbhrPHO060QbqZiHDo8MDx4SpLAUR26k5llaG4hzrTf98t4zHGTUEQrdRk9rrE8w45ihhc8JurZTBImYEJcT4jSR/4RuPUHmWXKEgarBu6St4UahCtUqxzLFvh+KcQC0MJOP4/gVja/q3W1USyTBACPER5MHKZE5Vcq1B1N3CzMS3U/hNkU+AH7RAPus1Dn1+5qCvWtxX3An/yqqg0WQVhaogS4p8hTwiwIeov8q4Bbs4hmL9RLxvGhdg3d80gP7a5XFGUSZqhNSqMEAQdBvpCdiT+ZP4ySjMeNk7hZBFx+gyuouBmLkqHOPO4BR1vUzlCn+XM11Fivgy+w4j3CKMkAb6nacLO4x6kVlbUz5geqMzqebPmMt8UWzgFcJmHSrRi4VQISaeDYunrtD9uteY5jQeeDScQDetvVInK+hLQdsUAi6DY2vsEXOCcxiVQfD1Z1qwSB+1f5Awx2LglEMbN0VbOVSs1AuiznFwv1igaMca1m0UvErfqAbOx0O9E2Xqs0xrZnJcD4aFHgV1lVTomEa8XzT+6W6eSGVePKM8vr43m07ASfc+KzR7nK1KwemtzKNQdP+pNbVtkfFRGX+pSXEoFHRFaXfyD5OvsfxE6uo/m4MLH3HzfV1qQTe50pGh72PX48sWGazeusxEsuJNl7vETwqDPHEutffvgeLzCuiz0lCmSf8pEtpfhTxnAf27sTxd0r3H3R/ynuw+rjlYFHpCicaB8virNlJ8/AXzhFZf9MePVIAAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ k – 2 = 0
⇒ k = 2
Question 12.In each of the following, find the value of k for which the given points are collinear.
(k, k), (2, 3) and (4, – 1)
Answer:(k, k) (2, 3) and (4, –1)
Area of triangle = ![](data:image/png;base64,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)
x1 = k, x2 = 2 and x3 = 4
y1 = k, y2 = 3 and y3 = –1
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANMAAAAgCAMAAAC/371cAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6Ojo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkGYAkGY6kLbbkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///brpB+jgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACvElEQVRYR+1Ya3fTMAyNBwUCpUAoj7AtwDAdJI3//89DluS8bMsuG+eUnPrD0sX2la50JbstistYSwTM1xdroeJ4/Hy9Xx2notAXTv+FUFeep/u9emfzYL5vf5yeD727kzc9DFarzU2mU6Z5/puXtuotfjQ1PXlodXWbBda9+iytQ9hssBHJwZpvivDNtVJbIX5awcDkAJVnTAn/xxml4F0rcjKQXbVB1t1LgbypEVYGgwX3HxAkAMsO2sexfpKlpL6iDtjS8sYGpSvhr+xGoz4Vx4r2aIpKcDCsB2bqaXaPe8XmA7ANTiFCyzlLCMjUyIkftFhblAQnu0uTCSFRDlbm1O3u0CSENADbYMiQmUtAglNBWzA1PPrK6sW6AUrcCBLmsLG+QoYcrAc2zxOHkRHmsBQWUmBfCZKY2KcITQOpsZLsG8CLy4pj56Iblh4XpQcmcSKxcdKAh40wcXLFD3NU9zACZc/qmHCavGm3EqUht5Tq0HCw9jkDEzgtYRF9yckzxgyxFkpq6qxn+5FTD24cBkqhsIySc5z8VQ42CTba92DNF1QMaS/rSufniT1sqaVHx6SrZORpCRbNkw876xF0ACW0xxkatdeVtA/edKUQlmYsNexWKe3NwaKcPFiqJ+qxmb3c63uuQrEE4ucBHkotmmOrIVLTvjcDi3HyYTlxtuWNLSIuHzuzPJ8GB9GFJnaZwDPJNMgp53xagnmcyFAIlpUN57LaSS1r5Lm4RwxNHcrp6tbU7gq1DAy3A5JF8h7hg804mesS6u3px6FMZrBOOHJuprOL+17+xmFlzn3vAbC5gvsFeXzDdiL38mwvcu7l2WDjQu9ensDo398Uh6FS1vL9KfEN4S/ieg5btDu9zsGZx/Hh4MrpceDOAUWvj1ILlFrxZ4RzCPxJPvSVvZmvi9NJAbgs/kcR+APwuVGFXKobLAAAAABJRU5ErkJggg==)
⇒ 7k – 2 – k – 12 = 0
⇒ 6k – 14 = 0
⇒ 6k = 14
⇒ ![](data:image/png;base64,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)
Question 13.Find the area of the quadrilateral whose vertices are
(6, 9), (7, 4), (4,2) and (3,7)
Answer:When the vertices of a quadrilateral is given then its area is given by
{(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
We must take all the vertices in counter clock wise direction otherwise it will give solution in negative.
So, from the figure we assume that
A (x1, y1) = (7,4)
B (x2, y2) = (6, 9)
C (x3, y3) = (3, 7)
D (x4, y4) = (4, 2)
![](data:image/jpeg;base64,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)
The area of the quadrilateral is
{(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
=
{(7–3)(9–2) – (6–4)(4–7)}
=
{28–(–6)} = 17 Sq. units
Question 14.Find the area of the quadrilateral whose vertices are
(–3, 4), (–5, – 6), (4, – 1) and (1, 2)
Answer:When the vertices of a quadrilateral is given then its area is given by
{(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
We must take all the vertices in counter clock wise direction otherwise it will give solution in negative.
So, from the figure we assume that
A (x1, y1) = (4, –1)
B (x2, y2) = (1, 2)
C (x3, y3) = (–3, 4)
D (x4, y4) = (–5, –6)
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fvil+TYenemzbP3fT74oAlwdw+Y9KAOT8xpvybh06UDZk9KAFx8h+Y96ZMPlj5P31/nTvk2Hp3pk2zbH0++v86AJcHcPmPSgD5j8xpvybh06UDZuPSgBcHafmPemTD5E5P31/mKd8m09O9Mm2bE6ffX+dAEpHzD5jQB8x+Y007Nw6UDZuPSgBQDtPzHvTJh+6Xk/eX+Ypw2bT070yXZ5a9PvL/MUASkfMPmNAHzH5j0pp2bh0oGzcelACgHB+Y9aZMP3I5P3l/mKcNmD061HNs8kdPvL/MUATEfMPmNGDuPzHpTTs3DpR8m49OlADgOvzHrRTRs56daKAG+a20fuZO3p/jTHlbz4v3Mnf09PrU3zbR07UyTd58XTv8AyoAPObcf3MnT2/xoErc/uZOvt/jUnzbj06Ug3YPTrQAzzW2j9zJ29P8AGmSSt50X7mTqfT0+tTfNsHTtTJN3nQ9Op/lQAec24/uZOnt/jQJW5/cydfb/ABqT5tx6dKQbuenWgDM127eHQborFIGaPYp46t8o7+9Ig8hLaFYZAIxtHA7Lj1pmukyf2facfvrlWI/2UG4/yFWZN3mxdPvH+RrZaQRk9ZMTzW3f6mTp6D/GlErc/uZOvoP8ad827t0oG7np1pDGea2z/UydPQf402SQ74v3Un3vQeh96k58vt0pJM+ZF0+9/Q0AHmtu/wBTJ09B/jQJWyf3UnX2/wAad827t0oG7J6daAM6fT4nc3Nuk9nc4/10GBu/3hnDfjQdWv7Ep/ado0sSt/x82657H7yZyPwrQ+by+3Skk3boun3/AOhqua+ktSeW2q0JrbUIbyMTW2ZoyPvIQf61KJmyf3MnX2/xrJn0qJ7g3Fu7WdyR/rYDjd/vDo340Jql/p+RqVuJ4h/y9Wyk4/3k6j8M1Lpp/CVz2+Il1vUrmw0x57dFWRAzBZQP3mATgfMOePyz1qPUrnzxpEqxPh7tGHA5yjVfjks9Usw6GC6h6jIDAH6HvVfWc+dpfT/j+X/0B6zaa3NE7l7zm3f6mTp7f40CZsn9zJ+n+NSfNu7dKBuyenWkBF5rbD+5k/T/ABps0rHy/wBzIPnHp/jU3zbD070ybd+76ffFAB5zbh+5k6e3+NAlbJ/cyfp/jUnzbh06UDdk9KAI/Nbaf3Mnf0/xpk0rFU/cyD5x6ev1qb5th6d6ZNu2x9Pvr/OgA85tw/cydPb/ABoErZP7mT9P8ak+bcOnSkG7celADPNbaf3Mnf0/xpk0rbF/cyffX09frU3zbT070ybdsTp99f5igA85tw/cyfp/jQJm3H9zJ+n+NSHduHSkG7celADBK2D+5k7+n+NMllby1/cyfeX09R71MN209O9Ml3eUvT7y/wAxQAGVsj9zJ+n+NHnNuP7mT9P8aed24dKUbtx6dKAIxK2D+5k6+3+NRyysYR+5k+8vp6j3qcbsHp1pk27yR0+8v8xQAGVsj9zJ+n+NHnNuP7mTp7f4087tw6Uvzbj06UARiVuf3MnX2/xoqQbuenWigBMHaPmPamSA+fFye/8AKnfJtHTt3pkm3z4unfv7UAS4O4/MelIAcH5j1o+TceR09aQbMHp19aAFwdg+Y9qZID50PJ6n+VO+TYOnbvTJNvnRdOp7+1AEuDuPzHpSAHn5j1o+TceR09aQbOenX1oAx7rM/iKFNxItbUufYuwH8gatSA+bFyfvH+RqlZlZtR1O54x5qwrz2RRn9SauybfNi6fePf2Nby0sjGOt2Pwd3U9KADzyetJ8u7t09aBt56dfWpKDB8vqelJIDvi5P3v6Gj5dnbp60km3fF0+96+xoAfg7up6UAHJ5PWk+Xd26etA25PTr60AGD5fU9KSUHdFyfv/ANDR8uzt09aSXbuj6ff9fY0APwd3U9KADk/MetJ8u7t09aBtyenX1oApTaTC7m5t5JLS5x/rYON3+8OjfjVLUr6/tJNP/tKNZYo7xW+0wDr8rdU6g/TNbPy7O3T1qlqm3zNN6f8AH8nf/Yeq5rq0tSeW22hrWt1Bexia2uFmjI+8hBqYA5PzHrWJPpdu85uLeRrO5x/rYDjP+8OjfjQmqXlgSNStxPEP+Xm2BOP95Oo/DNS6afwlKdviNrB2H5j3pkwP7vk/fFR211a3lt51tMkqEfeRs0+bb+76ffHes2mtzS9yXB3D5j0oAOT8xrjmKprszqYDMNRXCqG+0FflBwem317YzXXMQA5XBYDgZ71dSnyWM4T5rmba62l/qd7Z2aGaKwOy5nB4EmM+Wox8zAEZ9MgcnOMnSPGa694ov9CTTri1OnKju9wQGYlgMbRnAwc5zWb8HnabwRLPOczzX07yk9S5IzmqHhcD/hcvi3/ci/mlZmh6HqF39gs5boxzTmNSVihXc8h7KB6k4Fclofj+9vPF58Na3oEukXk0ZktszCUOACeSBjoDyPQiumttb0i9uruC21C3llsci5VZBmL13enQ/lXJaNf6HrfjY+JJtStGlaMWOl2qzK0pTJLSFQcgtlsDsvXrwAdVpesx6hc3unyKYL6xcLPCTnhhlXU91I/UEHpV+YHYnJ++v8xXCXMjQ/HS2SE/JPopEwB4IDsQT+IFd1Nt2J0++vf3oAlIO4fMaADuPzGmnZuHT86Bs3Hp+dACgHafmPemTA+UvzH7y/zFOGzaenfvTJdnlr0+8vf3FAEpB3D5jQAdx+Y9Kadm4dPzoGzcenT1oAUA4PzHrTJgfJHJ+8v8xThswenX1pk23yR0+8vf3FAEpB3D5jRg7j8x6U07Nw6fnS/JuPTp60AKAefmPWimjZz06+tFADPOO0fuZO3Yf4015j58R8qTjPYen1qb5to6dqZJu8+Lp3/lQAecdx/cydPQf40CY8/uZOvoP8ak+bcenSkG7B6daAI/OO0fuZPyH+NNkmPnRfupOCew9Km+bYOnamSbvOh6dT/KgA847j+5k6eg/wAaQ3HloztFIFXJJwP8al+bcenSs7XZng0O8ZcbmQov1b5R/OqiuaSQpOybKGiOw0eKRopC05aYnH95if5Yq9JKfMi/dSfePYeh96WGLyLWOFQAI0VR+AxSyZ82Lp94/wAjWsneTZnFWSQead3+ql6f3R/jQJTz+6l6/wB0f40/nd26UDPPTrUjI/NOz/VSdPQf40kkp3x/upPveg9D71Jz5fbpSSZ8yLp97+hoATzTu/1UvT+6P8aBKcn91L1/uj/Gn87u3SgZyenWgCPzTs/1UnT+6P8AGkklO6P91J9/0HofepPm8vt0pJc7oun3/wChoATzTu/1UvT+6P8AGgSnJ/dS9f7o/wAafzu7dKBnJ6daAM7VtQms7ISQIokZtqrIhYyNg4VQCMknH0GT2pNRmcnSy8Dq32yMsowcHY+QD3q5cWlveQhLq2hnVTlRKgYA+oyKr6mMPpoAAAvkAA7fI9IC35p3f6qXp/dH+NKJTk/upfyH+NO53dulA3ZPSmBnT6fC7m4t0uLO5/57QADP+8OjfjXKeJvGniHS9UXTIzbI0CLIZmhJMu7OPlzwOOcd/Su752Hp0rP1bRNM1ZoG1CwguWjcBWccgHqMjt7dK1hOKa51dGc4Nr3HZlnw/rMms6FZajJavG9xFuZV5XPQ49uMj2NaImOT+5k/If41hxO3hxwoXdpR4Cj/AJdPp/sfyreRt43KVZWAIIOQRWE42d1saxlfTqc/omjyaBfajDbQO2mX8zXSAEBreVhh168qcAjHTkehqSDw3o2n6m2qWmnSxXszjzZvNctJzn5stz0HWt35th6d6ZNu2x9Pvr/OoLOaj8G6S2q6peTaYqR6lH5csMWVEo3bmZyCOSccDjA75NS6Z4L8MaNqEeoadoKwXUOfLkXJK5BBxlvQmuk+bcOnSkG7celAHPaVpEq+Ib/xHqEDC8uo1t4Ilwfs8C/wk55Zj8x7DgDpk7cspKL+6k++vYev1qb5tp6d6ZNu2J0++v8AMUAHnHI/cyfkP8aBMdx/cyfkP8akO7cOlIN249KAGCY4P7mTv2H+NMlmPlr+6k+8vYeo96mG7aenemS7vKXp95f5igAMxyP3Mn5D/GjzjuP7mT8h/jTzu3DpSjduPTpQBGJjg/uZPyH+NRyykxAeVJ95ew9R71ON2D060ybd5I6feX+YoADMcj9zJ+Q/xo847j+5k6eg/wAaed24dKX5tx6dKAIxMef3MnX0H+NFPXdz060UAGDtHzHtTJAfPi+b1/lTvl2jkdu9Mk2+fFyO/f2oAlwdx+Y9KQA4PzHrR8m48jp60g2YPI6+tAC4OwfMe1MkB86H5u5/lTvl2Dkdu9Mk2+dFz3Pf2oAlwdx+Y9KyNc/ePYWmSfOugzD/AGUBY/yFa3ybjyOnrWNclZvEsSZytrbM55/ic4H6A1pS+K5FTaxdwdnWmSA+bFz/ABH+Rp3y7ByOnrTZNvmxcj7x7+xpiH4O7r2oAPPPek+Xd1HT1oG3nkdfWgAwfL69qSQHzIuf4v6Gj5dnUdPWkk274uR9719jQA/B3de1AByee9J8u7qOnrQNuTyOvrQAYPl9e1JKDui5/j/oaPl2dR09aSXbui5H3/X2NAD8Hd17elAByee9J8u7qOnrQNuTyOvrQAYOzr2qlqgPmabz/wAvyf8AoD1c+XZ1HT1qlqm3zNN5H/H8nf8A2HpAX8Hd17elAByef0pPl3dR09aBtyeR+dMAwdh57U2UH93z/wAtBTvl2HkdPWmS7f3fI/1g70ASFdx2nkEEEEdazY3k8PyEZZ9LY89zak/+yfyrR+XcOR09aMIdwOCDwQe9UnbR7Ca6ouKQ8QZXDKwyCMEEU2YHbH838a/zrCjkOgMeS+lueRnJtSf5p/KtuRo3jiZWDKXUgg5BFZyjy+hcZX9SbB3D5j0pADuPzGj5Nw5HT1oG3ceR+dQUGDtPzHvTJgdifMfvr/MU75dp5HfvTJtuxOR99e/vQBIQdw+Y0AHcfmNB2bhyPzoGzceR+dAAAdp+Y96ZMD5a/MfvL/MU4bNp5HfvTJdvlLyPvL39xQBIQdw+Y0YO4/MelB2bhyPzoG3ceR09aAAA4PzHrTJgfJHzH7y/zFOGzB5HX1pk23yRyPvL39xQBIQdw+Y0uDuPzHpSHZuHI/Oj5Nx5HT1oAUA8/MetFIuznkdfWigCPzvlH7qXt/DTHm/fRHypOM/w+1T/ADbRwO3emSbvPi4Hfv7UAHn/ADH91J/3zQJuv7qXr/dqT5tx4HT1pBuweB19aAI/O+UDypf++aZJN+9iPlScE/w+1T/NsHA7d6ZJu86HgdT39qADz/mP7qTp/drFspfO1TU7ny5DmZYVwvZF5/Umtx38sM7YAVcnmsTQg/8AZMcrD5p2aZv+BMT/AIVrD4W/6/rQzn8SRc835B+6l/75pJJf3kX7qT7x/h9jUnOwcDp602TPmxcD7x7+xoAPN+b/AFUvT+7QJev7qXr/AHaf827oOnrQN3PA6+tAEfm/J/qpen92kkl+eP8AdSfe/u+xqTny+g6etNkzvi4H3vX2NAB5vzf6qXp/doEvJ/dS9f7tP+bd0HT1oG7J4HX1oAj835P9VL0/u0kkvzR/upPv/wB32NSfN5fQdPWmy53RcD7/AK+xoAPN+b/VS9P7tAl5P7qXr/dp/wA27oOnrQN2TwOvrQBQ1TUJrHTJLiG1MjIVyH+UAFgCf16UmqyYm04eXJxfJ/D/ALL1edBJCUkRXUjlW5BqnqhbzdN6f8fyd/8AZekBb835v9VL0/u0CXk/upf++af827oOnrQN2TwPzpgR+b8h/dS/9802WX/V/upPvj+Gpedh4HT1psuf3fA/1g70AHm/MP3UvT+7QJeT+6l/75p/zbhwOnrQN2TwPzoAjMgMZBhkIIIIKcGsvzn0Fl/dytpbOCQV5tjnt6of0rY+baeB370yYblQMoILgEHvVJ20ewmuqLa3CvtZY3ZSMghcgilE3J/dS/8AfNYkcknh6XoX0tzyOptSf/ZP5Vuo28blKsrAEEHIIrOUeX0LjK/qM875T+6l7/w0yabKL+6k++v8PvUeqNOulzGCOR5OABExDY3ckEDPTPTn05qpo/2wWMi3ayAi6/debuzsyMfeJOOvU5o5fd5g5vesafncj91L/wB80Cbk/upf++akO7cOB+dIN248D86gojE3B/dS/wDfNNlmzGv7qT7y/wAPuKmG7aeB370ybd5S8D7y9/cUABm5H7qX/vmjzvmP7qT/AL5p53bhwPzpRu3HgdPWgCMTcH91L/3zUcs2YgPKk+8v8PuKnG7B4HX1pk27yRwPvL39xQAGbkfupf8Avmjz/mP7qTp/dp53bhwPzpfm3HgdPWgCMTdf3UvX+7RUi7ueB19aKAEwdo+b07UyQHz4vm9f5U75do59O9Mk2+fFz69/agCXB3H5u3pSAHB+bv6UfLuPPb1oG3B57+tABg7B83p2pkgPnQ/N3P8AKnfLsHPp3pkm3zoee57+1AFPX5Wg0W8ZW+Zo9i/VjtH86WCHyIEhU4EahRx6DFQa7tkksLTOfOuVZuf4UBY/yFWht557+tbLSCMnrJiYOwfN29KbID5sXzfxHt7GnfLsHPb1psm3zYuf4j39jSGPwd33u3pQAefm7+lHy7uvb1oG3nnv60AJg+X97t6U2QHfF838Xp7GnfL5fXt602Tbvi5/i9fY0APwd33u3pQAcn5u/pR8u7r29aBtyee/rQAmD5f3u3pTZQd0Xzfx+nsad8vl9e3rTZdu6Ln+P19jQA/B3fe7elAByfm7+lHy7uvb1oG3J57+tACYOz73b0qlqgPmab83/L8nb/Yervy7Ovb1qjqm3zNN5/5fk7/7L0gL+Du+929KADk/N+lHy7uvb1oG3J5/WmAmDsPzdvSmyg/u/m/5aDtTvl2Hnt602Xb+75/5aDvQA/B3D5u3pQAcn5v0o+XcOe3rQNuTz+tABg7T83r2pkoOE+b+Ne1O+XYefXvTZduE5/jXvQA8rngnIIIII61mxySeH5Dyz6Wx5HU2pP8A7J/KtL5dw57etJhG3A4IIwQT1qk7aPYTXVFxSHi3K4ZWGQRggimzA7E+b+Ne3uKwo5DoDHkvpbk5Gcm1J/mn8q25GjeKNlYMpZSCGyCMis5Q5fQuMr+pKQdw+b9KADuPzfpQdu4c/rQNu48/rUFAAdp+b17UyUHy1+b+Je3uKcNu08+vemS7fLXn+Je/uKAJCDuHzfpQAdx+bt6UHbuHP60DbuPPb1oAADg/N39KjmB8kfN/Evb3FPG3B57+tMm2+SOf4l7+4oAkIO4fN+lLg7j83b0pDtyOf1o+Xcee3rQAAHn5u/pRQNvPPf1ooAzf7cj2j/iXal/4CNTH1uMzRn+ztS4z/wAujVq/NtHA7d6ydX1KaKKVbeN4Jo1IS4mTEKk4GS3PHPpQBJ/bkeT/AMS7Uv8AwEaga5Hz/wAS7Uv/AAEapdGupLzTo5pG3vllZyRhirEZBAwQccHuKujdg8Dr60AZv9uR7QP7O1L/AMBGpj63GZYz/Z2pcE/8ujelavOwcDt3rOvtTa3lwthdytHnGyIlWOOMEf4UAZVxqyXHiGF/sN/ttrdjt+ytkMxxnH0Bq4NWTn/iX6l1/wCfNqo+F7u41OO51K52maVhExClAdmQcA8gZNbgzzwOvrW8tLIxj1ZQ/tZNg/4l+pf+AbU2TVk8yL/iX6l94/8ALo3oa0Odg4HT1qlfXpicpslg2bj9peLMSfKeSc8ioKD+1k3f8g/Uun/Pm1A1ZOf+JfqXX/nzajRrya+sFmm2mQO6EjGDtYgHjg8Y5HFXhnngdfWgCh/aybP+QfqXT/nzamyasm+P/iX6l97/AJ9G9DWhz5fQdPWqOo6h9mDblkgEYZjcyR5hTCk5Jz0FAC/2sm7/AJB+pdP+fNqBqyZP/Ev1Lr/z5tS6PdTXlgs8rpLuZtjqRkpn5dwHAbHUdqujOTwOvrQBQ/tZNn/IP1Lp/wA+bU2TVk3R/wDEv1L7/wDz6N6GtDny+g6etV5rlvtcUC205YvxJs/d52nq1AEX9rJu/wCQfqXT/nzagasmT/xL9S6/8+bU7Sbm4u9NhnufLMzbg5QYXIcjgfhVwZyeB19aAKH9rJs/5B+pdP8AnzaqWp6qhk0//QNQGL1DzasM/K/SrOt309jaxNFLHAHL7pXAYDCMwHPqQB/LmnXsjyx6RI8exnu4mZc/dJRiRQA/+1k3f8g/Uun/AD5tQNWTJ/4l+pf+AbVe53dB09aBnJ4H50AUP7WTYf8AiX6l/wCAbU2XVk/d/wDEv1L74/5dGrQ52HgdPWqOt3NzaacJ7byw6TR53gngsAcD15pgL/aybh/xL9S6f8+bUDVkyf8AiX6l/wCAbVfbIkxgcZ70gzk8D86QFD+1k2n/AIl+pd/+XNqbLqyYT/iX6l99f+XRq0edp4HfvWX4gvnsLGKRLmK3keULHvAJkbBwo3EAcjJJ6AGgCb+1k3D/AIl+pf8AgG1A1ZMn/iX6l/4BtVyMuUjL7CxQFthyucDOPanDOTwPzoAof2rGUKnTtRIOQQbNuazY9UOitgWWof2WWDEPbMDbHPb1U+nauhGdp4HfvWbr9zLa6U0kU0UT7htEm3DnBO35sAZx37Zqk7aPYTXVFsa9E4Vl0/USpGQRaNg0v9uR5P8AxLtS/wDARqzrO5k0WKGYqW0ydFcrg5tSwB6HnZz07V0SPv8AnUqysAQQcgiolHl16Fxlf1M7+3I8H/iXal/4CNTJdbjMYH9nal95f+XRvUVqjdtPA796yvEuoXOm6HLcWsJeYD5GEbOqEDO5gOccfnioKHnXI8j/AIl2pf8AgI1H9uR5P/Eu1L/wEar8MjSwwyYxvQNzweR6VIN248Dp60AZg1yPB/4l2pf+AjUyXW4zEB/Z2pdV/wCXRvUVqjdg8Dr61l+Ibq5tdKRreeC3kaaNfMmbCqM88kEA8Y5oAcdcjyP+JdqX/gI1H9uR5/5B2pf+AjVasLhruwtbgqwMsSvhxhuRnkCrPzbjwOnrQBmDXI+f+JdqXX/n0aitNd3PA6+tFACYO0fN6dqjkXM0QJyDu7e1P+XaOfTvTJNvnxc+vf2oAkC4OAcAAYAFABwfm7+lHy7jz29aBtwee/rQAYOwfN6dqinby3jdm+VdxP5Go7q/s7JY/tNwsW/7u4nt1P09+lVfEUwh0iZlPzMjIvPdhgfzqoxbaXclyST8itoSt/ZUUrHDTlpjx/eYn/CtAA889/SmQxJBFHCpwI0VRz6DFPG3nnv61pJ3k2RFWSQmDsHPb0psgPmxc/xHt7GnfLsHPb1psm3zYuf4j39jSGPC4IAwABwAOlAB557+lGF3de3rQNvPPf1oATB2de3pTZVy0YOCC3Qj2NOwuzr29abJt3xc/wAXr7GgByoE+VAqjHRVAFKAcnnv6UYXd17etAC5PPf1oATB8vr29KSUHdFz/H6expfl8vr29abLt3Rc/wAfr7GgB+CG69vSgA5PPf0o+Xd17etAC5PPf1oAQrmPBwR6EVS1QHzNN+b/AJfk7f7D1d+XZ17etUtU2+ZpvP8Ay/J3/wBh6QF7B3de3pQAcn5v0owu7r29aBtyef1pgJg7Dz29KbKCfLyc/vB2p3y7Dz29abLt/d8/8tB3oAfg7h83b0oAOTz+lGF3Dnt60Dbk8/rQAYOw8+vao541dUDhXHmLwyg0/wCXaefXvTZduE5/jXvQA/BBHPb0oAOTz+lGF3Dnt60Dbk8/rQAYO08+vamTLujUNgjevBHuKcAu08+vemy7dic/xr396AHsN3DEEEEEEdazY3k8PyHln0tjyMZNqT/7J/KtIhcjn9aTCncDggjBBPWqTto9hNdUXEO+MMrhlYZBHIIpswPlLk8bl7e4rDjkbw+55Z9LdjkA5NqT390/lW3IyPAjK4ZWZCCGyCMis5w5fQuMr+pIQ2Rz+lAB3H5u3pXOalqE8esm8jSdrTT2WOVkYeXhv9YWGcnaCh6HGDXRjbuPPb1olBxSfcUZKTYAHB+bv6VHOu6ABsEErwR7ipBtwee/rUc23yRz/Evf3FQWSEHcPm/Slwdx+bt6Uh27hz+tHy7jz29aAAA8/N39KKBt557+tFAEfnLtH7uTt/yzNMeZfPiOyTjP8B9Knydo49O9MkJ8+Lj17+1AB5y7j+7k6f8APM0CZef3cnX/AJ5mpMnceO3rSAnB47+tAGbeW0k1wtzbStBJ5JhbdblxtJByB6jHfI9qra7Is0umWu2T57oM2UPKqCT/AErbydg49O9Yt45m8VW0eMi3tnY89Cxx/JTW1Ntv0M5pJepb81dx/dydP+eZoEq8/u5Ov/PM0/J3Hjt60Annjv60gI/NXYP3cn/fBpJJV8yL5H+8f4D6GpMnYOO3rTZM+bFx/Ee/saADzV3f6uTp/wA8zQJV5/dydf8Anmafk7unb1oBPPHf1oAj81dn+rk6f3DSSSrvj+ST739w+hqTJ8vp29aSTPmRcfxevsaAE81d3+rk6f8APM0CVcn93J1/55mn5O7p29aATk8d/WgCPzV2f6uTp/cNJJKu6P5JPv8A9w+hqTJ8vp29aSXO6Lj+P19jQAnmru/1cnT/AJ5mgSrk/u5P+/Zp+Tu6dvWgE5PHf1oAyrvXoE0+WeyQ3TxFd64KBQWxkkj9BUuqyKJtOGyTi+T+A/3Xq9KgngaOZBIjDlW5BqnqpJl03j/l+Tv/ALL0gLfmru/1cnT/AJ5mgSrk/u5P+/Zp+Tu6dvWgE5PH60wI/NXYf3cn/fBpssq/u/kk++P4DUuTsPHb1psuf3fH/LQd6ADzV3D93J0/55mgSrk/u5P+/Zp+TuHHb1oBOTx+tADPNXaf3cnf/lmaZLKuE+R/vr/Aamydh49e9MlzhOP4170AHmruH7uT/v2aBKuT+7k/79mn5O4cdvWgE5PH60AM81dp/dyd/wDlmaZLKu1fkf76/wAB9amBO08evemS52Jx/Gvf3oADKuR+7k/79mjzVyf3cn/fs08k5HH61DdXkFlGZbmRY16DJ5J9AOpNCTewXsO8xCrAxuQc5BjPNcT4g8TXvha/XS9LWJoXjW423MbHycsflUAjg7SfbtXVp/ampj9yh062P/LWQZmcey9F/HmuU+I2maXZ6XpdlHak3+pX8drHdlyZUDEb23d+OMHjnOK1jKEH7+q7GclKS9zR9zrvDeurr2h2+oPbGJ5N6yIill3KxVsHuMitOO6hkLeXl9p2ttXO0jqDjofasu8xptrb6HokSW8rRkR7FyttEOshHc9gD1Y+xrmvgtJJL4HklkZpJJL6ZndzksTtySe5rmdr6G621O7Ey4P7uTr/AM8zUVxcRpBucMigrksuB1FStPHGSryRqfQuBXmvxeg2T+HLgSzZl1FEaPzm8sgYI+Xpn3xSGekm4j3hdr7gM42HOKXzl3H93J0/55mua+IUk+neHJNesiI77SnWaJ/767gHRvVWUnj1weoFdBp18mpafbX8K/u7mBJUyezDI/nQBKJl5/dydf8AnmaKkBPPHf1ooATB2jn07UyQHz4ufXt7U7A2jn070yTHnxc+vf2oAlwdx57elIAcHnv6UYXcee3rQNuDz39aADB2Dn07VhWpM2s31zn/AJb+Spx2ROf1JrbZkSIuzYCjJ59KwtGX/iXWsrH5p2kmbn+9k/yxWsPhb/r+tDOfxJGng7jz29KADzz39KMDcee3rQAOee/rQAmDsHPb0psgPmxc/wAR7exp2BsHPb1psgHmxc/xHv7GgB+Du69vSgA889/SjA3de3rQAOee/rQAmD5fXt6UkgO+Ln+L09jS4Gzr29abIBvi5/i9fY0APwd3Xt6UAHJ57+lGBu69vWgAZPPf1oATB8vr29KSUHdFz/H6expcDy+vb1psoG6Ln+P19jQA/B3de3pQAcnnv6UYG7r29aABk89/WgBMHZ17elUtUB8zTef+X5O3+w9XcDZ17etUtUA8zTef+X5O/wDsPQBewd3Xt6UAHJ5/SjA3de3rQAMnn9aAEwdh57elNlB/d8/8tB2p2BsPPb1psoH7vn/loO9AD8HcOe3pQAcnn9KMDcOe3rQAMnn9aADB2nn17UyUHCc/xr2p2BtPPr3psoGE5/jXvQA/B3Dnt6UAHJ5/SjA3Dnt60fLljuwB15oAADtPPr2qK5kSGASyyLGispZm4A5qmdSa5ZoNLhN3ICQ0pbbCn1bv9BUqaIrNHcalOb2YOu1TxFHyPur/AFOaqyj8RN2/hI1u73UmC6XEFi73c64X/gK9W/lVyy0WC1nNxK7XV1jmebkj/dHRR9K0MLkc/rQNu48/rUOo9loi1BbvUADtPPr2rivibp0sum6VrUYLjRdQiuplUZPlAjcQPbg/TNdqNu08+veo5gpiAznLLnn3FZlnP6/4H0PxTeJqV4bh5TAI0eC4ZFZQSR04P3jzXLeHPCHiLwV4E1H7DcWdrrMswd5JpS8KxL6ZGAcFu39K9It7W3s4kgtkWKJc7UU4C+wHYe1PeOKTKyKrgYOG5GR0oA5WLwdovimxs9Y8R6JC+qXFtGbjLOuG2jjAbiqXxC8Oavr/APY8el2sRi065WdmkmCbgMAKBz+tdyNuDz39ajmx5I5/iXv7igDmPiM80/gW5s44SbzUTHbW9uCCzSMw4/AAn6A1v6Np50rR7LTg4b7JbRw7sddq4z+lTvaWz3kd00atPGpVHPJUHrj0zUvy7jz29aAFAPPPf0opBt557+tFAEfmptHyP2/5Zn/CmPKnnxfI/f8A5Zn0+lT5O0cenemSE+fFx69/agA85Nx+R+n/ADzP+FAmTB+R+v8AzzP+FSZO48dvWkBODx39aAMzXroRaDdFFcO0exfkI5bgfzpqeXAttCqOBGu3/VnsuPSk11jIdOtMf665VmGf4UG4/wAhViQnzouO57+1bLSCMnrJieam4/K/T/nmf8KBKnPyv1/55n/Cn5O48dvWgE88d/WkMj81Nn3X/wC/Z/wpskqeZF8r/eP8B9DUuTsHHb1pshPmxcfxHv7GgA81N33X6f8APM/4UCVOflfr/wA8z/hT8nd07etAJ547+tAEfmps+6/T/nmf8KbJKm+P5X+9/wA8z6Gpcny+nb1pJCd8XH8Xr7GgBPNTd91+n/PM/wCFAlTJ+V+v/PM/4U/J3dO3rQCcnjv60AR+amz7r9P+eZ/wpskqbo/lf7/9w+hqXJ8vp29aSUndFx/H6+xoATzU3fdfp/zzP+FAlTJ+V/8Av2f8Kfk7unb1oBOTx39aAMm612BbN5LAxXLR/eVtw65wAACSxOAB71JqMwY6YzRSIxvYyVKHKnY/FWryyttRhRby3WYRNvTcx+VsEZGPYn86g1MbX0xQuAL5AOf9h6QFzzU3fdfp/wA8z/hQJUyflf8A79n/AAp+Tu6dvWgE5PH60wI/NTZ91/8Av2f8KbLKn7v5X++P4DUuTsPHb1pspP7vj/loO9AB5qbh8r9P+eZ/woEqZPyv/wB+z/hT8ncOO3rQCcnj9aAI/NTaflfv/wAsz/hTZZVIT5H++v8AyzP+FR3epW9kBHIS8z52Qxjc7fQCoHs9S1FUN4xsbYuP3ETZkb/eft9BVcul3oib62Q641eBJzb20Mt3dAcwxJ93/ePQUiaZLfNv1eRmTqLSBGEY/wB44y38q1rW0gsY1htbdIox2Xv7n1qYE7jx+tS6lvhKUL/EQxtDFCI44SiKMKqxEAfpSSypsT5X++v/ACzPr9Knydp49e9MmJ2Jx/Gvf3FZGgGZNw+R/wDv2f8ACgTJuPyP/wB+z/hUhJ3Dj9aQE7jx+tADBKm0/I/f/lmf8KZLKnlr8j/eX/lmfUe1TAnaePXvTJSfKXj+Je/uKAAzJuHyP/37P+FHnJuPyP8A9+z/AIU8k7hx+tKCdx47etAEYmTB+R+v/PM/4VHLKnkj5H6r/wAsz6j2qcE4PHf1pkxPkjj+Je/uKAAzJkfI/wD37P8AhR5ybj8j9P8Anmf8KeSdw4/Wlydx47etAEYmTn5H6/8APM/4UVICeeO/rRQAmG2jn07UyQHz4ufXt7U/A2jk9u9RyAefFye/f2oAlw248jp6UgDYPPf0owNx5PT1oAHPJ6+tAGNdEz+I4VzkWtqWPHQuwH8gatPnzYue57e1U7LE2panc5485YVOeyKM/qTVxwPNi57nv7VvLSyMY9WPwdx57elAB557+lGBuPPb1oAHPJ6+tSUJg7Bz29KbJnzYuf4j29jTsDYOT09abIB5sXP8R7+xoAfg7uvb0oAPPPf0owN3Xt60ADnk9fWgBMHy+vb0pJM+ZFz/ABensaXA8vr29abIBvi5/i9fY0APwd3Xt6UAHJ57+lGBu69vWgAZPPf1oATB8vr29KSXO6Ln+P09jS4Hl9T09abKBui5/j9fY0APwd3Xt6UAHJ57+lGBu6np60ADJ57+tACYOzr29KpapnzNN5/5fk7f7D1dwNnU9PWqWqAeZpvP/L8nf/ZegC9g7uvb0oAOTz+lGBu69vWgAZPJ/OgBMHYee3pTZc/u+f8AloO1EkkcMDSSyiNFByzNgCs9ru71LYNMiKQ7x/pc4IX/AICvVvr0pqLeom0i7d3cFkgluZljXoM9WPoB1JqtGNU1QnyQdOtj/wAtJFzM49l6L+PNW7LRba1nFxK73V0RzPMcsPoOij6VoADJ5P50OcY/DqCi3uVLHS7XT42Nun7xs75X+Z3+rGrEwO2Pn+Ne3vTsDYeT370yYDbHyfvr396ybbd2aJJKyJcNuHI6elIA248/pRgbhyenrQANx5P50hhhtp59e1MmB2Jz/Gvb3FPwNp5PfvUcwGxOT99e/uKAJCG3DkflQA248j8qCBuHJ/OgAbjyfzoAAG2nn17UyYHy15/iXt7ingDaeT370yUDyl5P3l7+4oAeQ24cj8qAG3HkdPSggbhyfzoAG48np60AADYPPf0pkwPkjn+Je3uKeAMHk9fWo5gPJHJ+8vf3FAEhDbhyPypcNuPI6elIQNw5P50YG48np60AADc8jr6UUADn5j19aKAI/Oj2j5W7f8sz/hTHlj8+L5W7/wDLM+n0qfJ2j5fTvTJCfPi49e/tQAedFuPyt0/55n/Cmm4hRGdgwC5J/dn/AAqbJ3H5e3rWdr07QaFeMowzIUXnu3yj+dVFc0khSdk2UNEdRo8Ujq2+ctM3yH+Js+npirryx+bFw3U/wH0+lPhj8i2ihVcCNFUc+gAocnzYuO57+1ayd5NmcVZJB5se4/K3T/nmf8KBLHz8rdf+eZ/wp+TuPHb1oBPPHf1qRkfmx7B8rf8Afs/4U2SWPzIvlb7x/gPofapcnYOO3rSSE+bFx/Ee/saAE82Pd91un/PM/wCFAlj5+Vuv/PM/4U/J3dO3rQCeeO/rQBH5sez7rdP+eZ/wpskse+P5W+9/cPofapcnZ07etJIT5kXH8Xr7GgBPNj3fdbp/zzP+FAljyflbr/zzP+FPyd3Tt60AnJ47+tAEfmx7Put0/wCeZ/wpskse6P5W+/8A3D6H2qXJ8vp29aSUndFx/H6+xoATzY933W6f88z/AIUCWPJ+Vv8Av2f8Kfk7unb1oBOTx39aAMq61y3WxmnsPLvGgUvIqlsqoB9ATnjH6npTtQuIpP7LkUEq15Gw+Qngo/51durSC+gEdzAJVByAT0PI7exIrL1jUYkuLC1tU+0XEd2hEMRGBhWGCei//WppN7CbS3NczRg5IYADk7D/AIVQOrfaXaLSrc3bg4aUqREn1bHP0FTJo9zfMH1eUMnUWkJIjH+8erfyrWiRYU8uONUReFVcAChuMfN/gCUn5GRb6TCzC41KV72deVUxERRn/ZXH6nNaU0sf7vCtw4/5Zn/Cp8nYfl9e9MmJ/d8fxjvWcpOW5aio7B50W4fK3T/nmf8ACgTR5Pyt/wB+z/hUmTuHy9vWkBOT8v61JRH50ew/K3f/AJZn/CmTSx7U+Vvvj/lmf8Knydh+X170yYnanH8a9/egA86LcPlbp/zzP+FAmjyflb/v2f8ACpMncPl7etICdx+X9aAI/Oj2n5W7/wDLM/4UyWWPYnyt99f+WZ9fpU+TtPy+vemTE7E4/jXv7igAM0e4fK3/AH7P+FAmj3H5W/79n/CnkncPl/WgE7j8v60ARiaPB+Vu/wDyzP8AhTJZY/LX5W+8v/LM+o9qnBO0/L696ZKT5a/L/Evf3FAAZo9w+Vv+/Z/wo86Pcflb/v2f8KeSdw+X9aATuPy9vWgCMTR4Pyt1/wCeZ/wpkssfkj5W6r/yzPqPapwTg/L39ajmJ8kcfxL39xQApmjyPlb/AL9n/Cjzotx+Vun/ADzP+FPJOR8v60uTuPy9vWgCMTR8/K3X/nmf8KKeCefl7+tFAB820cjt2pkmfPi5Hft7U/A2jk9u9MkH7+Lk9+/tQBJ8248jp6Vka5mV7C0znzroM3H8KAsf5CtbA3Hk9PWse5/feJY0ySLW2Zjz0Ltj+QNaUviv2IqbWLh3bRyO3amPnzYuR1Pb2p2PkHJ7d6a4/excnqe/tTEP53HkdPSgZ55HX0ox8x5PT1oA68nr60AJzsHI6elNkz5sXI+8e3sadj5ByenrTZB+9i5P3j39jQA/nd1HT0oGeeR19KMfN1PT1oA68nr60AJz5fUdPSmyZ3xcj73p7GnY/d9T09abIP3kXJ+96+xoAfzu6jp6UDOTyOvpRj5up6etAHJ5PX1oATny+o6elNlzui5H3/T2NOx+76np61Xv7q3skjluJhGu/jJ5PB6DqaEm9EDdizzu6jp6VVu9RgsnEcjF5nPyQRrudvoP61Ci6nqjDyg2nWxH+skGZnHsvRfx5rSsNLs9PD+Qh3sfnldtzv8AVjzVPljuSry2M5bHUtRj3Xsv2G2I/wBRE37xh/tP2+gp99Z29iNLhtYUiQXy8KOvyP19areInnbzYDJcLEYU8lIo9wlbcdwY4PQBfTqTV/WR++0zn/l+Xv8A7D0pt8q7McLXZp/Nu6jp6UDdk8jr6UmBu6np60ADJ5PX1rE1D5th5HftTJs/u+R98dqdgbDye/emTD/V8n7470AS/NuHI6elA3ZPI/KkwNw5PT1oAGTyfzoAPm2Hkd+1Mmztj5H317e9Px8h5PfvUcw+WPk/fXv70AS/NuHI6elIN248j8qMDcOT09aABuPJ/OgA+baeR37UybOxOR99e3uKfgbTye/emTD5E5P317+4oAkO7cOR+VIN248j8qCBuHJ/OgAbjyfzoABu2nkd+1Mlz5S8j7y9vcU8AbTye/emSgeWvJ+8vf3FADzu3DkflSjduPI6elIQNw5P50ADceT09aAAbsHkdfSmTZ8kcj7y9vcU8AYPJ6+tRzD9yOT95e/uKAJDu3DkflS/NuPI6elIQNw5P50YG48np60AKu7nkdfSikAHPJ6+tFAEfnQ7Rwe38B/wpjzRedFwe/8AAfT6VPk7R8vp3pkhPnxcev8AKgBPOh3Hg9P7h/wrFspo5dT1O5OcGYRL8h6IvPb1JrdeTyw7sMKq5PNYuhbv7IjlYfNOzTNz/eYn+WK1h8Lf9f1oZz+JIt+dFtHXt/Af8Ka8sXmxdep/gPp9Kmydg49O9NcnzYuO57+1ACedFuPXp/cP+FAmi569f7h/wp+TuPHb1oBPPHf1oAj86LYOv/fB/wAKbJLF5kXX7x/gPofapsnYOO3rTZCfNi4/iPf2NACedFu79P7h/wAKBNFz16/3D/hT8nd07etAJ547+tAEfnRbO/T+4f8ACmySxb4+v3v7h9D7VLk+X07etNlYh4yRgBj39jQAedFu79P7h/wpj3VtBG8s0gjRerMpAH6VVOqSXUrQ6Vb/AGtxw0pOIU+rd/oKsW+iB5RcalIb2dTlVYYijP8Asr/U81XKo/ETdv4SmL+41BdumweXCRzdzxnb/wAAXGW/lVq202ztJ47iRpLm6LfNPMpLdD0GMAfStXJ8v7vamyk7ovl/j9fY1DqO1loi1BbvUTzot3fp/cP+FAmhyeD/AN8H/CpMnd93t60oJyfl7+tZlmHqXivS7DT5J4biG6kUfLFHIAW7nnsAAT+FS6xNEZdLK5x9uU52n+49aFzaW95atDc2yTRsCCrgEGqescS6WAuAL5f/AEB6AL3nQ7uh6f3D/hQJocng/wDfB/wqXJ3fd7etAJyfl7+tAEXnRbDwf++D/hTJpoj5eAfvj+A/4VPk7D8vr3pkxP7vj+Md6AE86HcOD0/uH/CgTQ5PB/74P+FS5O4fL29aATk/L+tAEXnRbTwe/wDAf8KZNNEVTGfvj+A+v0qfJ2H5fXvTJidqcfxr396AE86HcOD0/uH/AAoE0OTwf++D/hUuTuHy9vWkBO4/L+tAEfnQ7Twe/wDAf8KZLNEUXg/fX+A+v0qfJ2n5fXvTJidicfxr39xQAhmh3Dg/98H/AAoE0O48H/vg/wCFSkncPl/WkBO4/L+tAEYmhweD3/gP+FMlmi8teD95f4D6j2qcE7T8vr3pkpPlrx/Evf3FACGaHcOD/wB8H/Cjzodx4P8A3wf8KkJO4fL+tLk7j8vb1oAiE0ODwev9w/4UyWaIwjg9V/gPqPapwTg/L39aZMT5I4/iXv7igBDNDkcH/vg/4UedDuPB6f3D/hUhJyPl/Wlydx+Xt60ARCaHng9f7h/woqUE8/L39aKAE+baOR27UyTPnxcjv29qfj5Rye3emSD9/Fye/f2oApa/M8Gi3jKRuaPy147sdo/nSwReRbpCuAI1Cjj0GKg10eZLp9pknzrlWYf7KAsf5CrQHXk9fWtlpBGT1kw+baOR27Ux8+bFyOp7e1Px8g5PbvTHH72Lk9T39qQx/wA248jp6UDPPI6+lGPmPJ6etAHXk9fWgBOdg5HT0psmfNi5H3j29jTsfIOT09abIP3sXJ+8e/saAH/Nu6jp6UDdk9OvpVW71C3spFjd3eZx8kMY3O30H9ajjsNR1LJvZWsrY/8ALvC+ZGH+0/b6D86rl0u9ETfWyCfU0R/stqjXl1j/AFUX8P8AvN0Wg6PcXrxPq8yuhbi0hyIxwfvHq38q07WztrK1ENrEsMY7KOv19aklHzxcn7/r7GpdS3wlKF/iFjjWILHGqIijCqq4A/Cnjdk8jr6UmPm6np60Acnk9fWsjQT5vL6jp6U2Xdui5H3/AE9jTsfu+p6etNlHzRcn7/r7GgCT5t3UdPSgbsnkdfSkx83U9PWgDk8nr60AHzeX1HT0rO1nPnaX0/4/l/8AQHrQx8nU9PWs7WR++0vk/wDH8v8A6A9AGn827qOnpQN2TyOvpSY+bqenrQByeT19aAD5th5HftTJs/u+R98dqfj5Dye/emTD/V8n7470ASfNuHI6elA3ZPI/Kkx8w5PT1oA5PJ/OgA+bYeR37Uybdtj5H317e9Px8h5PfvTJh8qcn769/egCT5tw5HT0oG7ceR+VJj5hyenrQB8x5P50AHzbTyO/amTbticj769vcU/Hynk9+9MmHyJyfvr39xQBId24cj8qQbtx5H5UEfMOT+dAHzHk/nQADdtPI79qZNu8peR95e3uKeB8p5PfvTJR+7Xk/eXv7igCQ7tw5H5UDduPI6elIR8w5P50AfMeT09aAAbsHkdfSmTbvJHI+8vb3FPA4PJ6+tRzD9yOT95e/uKAJTu3DkflR8248jp6UhHzDk/nRj5jyenrQAq7ueR19KKQDryevrRQBH58O0fh/Cf8KY80PnRH0z/CfSp8naPlPamSE+fFwe/8qAMe5mim8TRDPy21qW+6fvO2P5A1bE0PPPf+6f8ACqtixm1XVLnHHnLCv0Ref1Jq8CcHg9fWt5aWRjHW7I/Oh2jn0/hP+FNeWHzYue5/hPpU2TsHB7d6a5PmxcHqe/tUlDfOh3Hnt/dP+FAmh557/wB0/wCFOlmSBGlmcRxqOWZgAKopeX2pZXS4fLhJ5u5xhf8AgK9W+p4qlFvUTaRNc6hZWkAe4mWMHplTlj6AY5NVDJe6nJH5atp1tuP7x0zM3B6L0X8ea0bLR4LRvtEm+6uiOZ5iCw+g6KPpV2QnzYuD94/yNJzjH4QUW9ynYWenadu+zod7D55XBZ3+pNWxPDz9f7p/wqXJ3fdPSgE8/KetZNtu7NEklZEPnQ7Ovb+6f8KbJNCXi/3v7p9DU+T5fQ9KbITvi4P3/wChpDG+fDu/D+6f8KBPDz9f7pqXJ3fdPSgE5PynrQBD50Ozr2/un/CmyzQlov8Af/un0NT5Pl9D0pkpO6Lg/f8A6GgBPPh3fh/dP+FAnhyf/iTUuTu+6elAJyflPWgDGv8AxDawWgexEd+27DiJiwjXBJZtqsQOPTvRql1BKdJkRgyteIwKgkEFHxjir11p9pewBLi3DKCSNrFT0IPIIPQkfjVbV1CPpSKgVVvUAA4AGx6AL/nw7vw/un/CgTw5P/xJqXJ3fdPSgE5PynrQBD50Ow8/+On/AAps00J8vH98fwmp8nYeD3pkxP7vg/fFACefDuH0/umgTw5P/wASf8Klydw+U9KATk/KaAIfPh2nn1/hP+FNmmhKpj++P4TU+TsPB70yYnanB++v86AE8+HcPp/dNAnhyf8A4k/4VLk7h8p6UAncflNAEXnw7T+P8J/wpks0JRf99f4T61Pk7T8p70yYnYnB++v8xQAhnh3D/wCJP+FAnh3H/wCJP+FSkncPlNAJ3H5TQBEJ4cH8f4T/AIUyWaExr/vL/CfUVOCdp+U96ZMT5a8H7y/zFACGeHcP/iT/AIUefDuP/wASalJO4fKaMncflPSgCITw4PPf+6f8KjlmhMIHuv8ACfUVYBOD8p60yYnyRwfvL/MUAIZ4cj/4k/4UefDuP0/umpSTkfKaMncflPSgCITw8/X+6f8ACipQTz8p60UAJ820dO1RTv5ciOxAChifyqXHyj5j2rN8QSGHS5SpO50aNfq2FH86qKu0hSdlcq6Erf2VFK2N05aZvqzE/wCFaAzg9Otc9p3nW1jp8i3s8kslwIGheTcrrvZThe2AM8elak+pxpM1rapJeXWf9VEeF/3m6LXRODcnYwhJcquXDkJkkAAck1nSalJd3CQ6VCLp1YhpTkQocd27/QVKmiz3oEmrz704ItYSRGP949W/lWp5SRPBHGoRFJAVRgDis24x83+BaUpeRQt9DDzC41Kb7bOvKhhiOP8A3V/qea1F3c9OtLj5j8x6UgHX5j1rOUnLctRS2D5tg6dKZJu82Hp94/yNOx8g+Y9KbIP3sPJ+8f5GpKJPm3dulIu7np1ox833j0oA6/MetACfN5fbpTZd3mRdPv8A9DTsfu+p6U2UfPFyfv8A9DQA/wCbd26Uo3ZPTrSY+b7x6UAcn5j1oAT5vL7dKbLu3RdPv/0NOx+7+8elNlHzRcn7/wDQ0AP+bd26Uo3ZPTrSY+b7x6UoHJ+Y9aAE+by+3Ss7Wc+dpfT/AI/l/wDQHrRx+7+8elZusj99pfJ/4/l/9AegDT+bd26UDdk9OtJj5vvHpQByfmPWgA+bYenemTbv3fT74p+PkPzHvTJh/q+T98UASfNuHTpQN2T0ox8w+Y9KQDk/MaAD5th6d6ZNu2x9Pvr/ADp+PkPJ70yYfLHyfvr/ADoAk+bcOnSkG7celLj5h8x6UgHzH5jQAfNtPTvTJt2xOn31/mKfj5T8x70yYfInJ++v8xQBId24dKQbtx6UEfMPmNAHzH5jQADdtPTvTJd3lL0+8v8AMU8D5T8x70yUfu15P3l/mKAHnduHSlG7cenSkI+YfMaAPmPzHpQADdg9OtMm3eSOn3l/mKeBwfmPWo5h+5HJ+8v8xQBId24dKX5tx6dKQj5hyaXHzH5j0oARd3PTrRQB1+Y9aKAIvPg2j5h27GvP/FvjlItX+wW9gJrewnVp3Mm1pCoyVUY7ZHJ6kfjXouTtHyntXNax4H0TWNYW+uYZ1km/1yRy7UmwP4h9MDjGRW1GUIyvNaGVWM5RtEZZWEWqbL5lhsLe4QOqWyASyK3PzOBxnPIH51vWkdhZQeTbRpFGOyr/AD9asoqxgIke1FUBVGAAKUE4PynrUSqSluVGCjsRefBtHzDt2NNkng82I7hwT29qnydg4PamSE+dDwep/lUFiefBuPzDp6UCeDn5h19DUuTuPynpSAnn5T1oAi8+DYPmH5GmyTwGSL5hwx7exqfJ2Dg0yQnzYuD94/yNACefBu+8OnoaBPBz8w6+hqXJ3fdPSgE8/KetAEPnwbPvDp6GmyTwF4vmHDensanyfL6HpTZSd8XB+/8A0NADfPg3feHT0NAng5+YdfQ1Lk7vunpQCcn5T1oAh8+DZ94dPQ02WeAtF8w4f09jU+T5fQ9KZKTui4P3/wChoATz4N33h09DQJ4Mn5h19Klyd33T0oBOT8p60AYupeJtOsoE8iSG7kckBIpM4wpYk4BxwD2/xp2qXMEp0qRGG1rxG45/gerlzptndwBJbYYBJBQ7D0IPKkHkEj8ar6sqxtpSIm1VvUCgdANj0AX/AD4N33h09DQJ4Mn5h+RqXJ3fdPSgE5PynrQBD58Gw/MPyNMmngPl4YffHarGTsPB70yYn93wfvigBPPg3D5h09KBPBk/MPyNS5O4fKelAJyflNAEPnwbT8w79jTJpoCEww++O1WMnYeD3pkxO1OD99f50AJ58G4fMOnpQJ4Mn5h+RqXJ3D5T0oBO4/KaAIfPg2n5h37GmSzQFFww++vb3qxk7T8p70yYnYnB++v8xQAnnwZHzD8jQJ4Nx+YfkalJO4fKaATuPymgCETwYPzDv2NNlmgMa/MPvL29xU4J2n5T3pkpPlrwfvL/ADFACGeDI+YfkaPPg3H5h+RqUk7h8poBO4/KaAIhPBg/MOvoajlmgMQAYdV7e4qwCcH5T1pkxPkjg/eX+YoAQzwZHzD8jR58G4/MOnpUpJyPlNGTuPynpQBEJ4OfmHX0NFSgnn5T1ooAT5to6dq5zxZrk+nXWk6XYsi6hqtx5MTkZ8lBzJJjuQOgPGTXR4+UfMe1cJ4ztpIfiP4M1Jyfs4mltyx6B2X5R+PP5UAdxDEYY1iDs4RQN0jFmPuT615V4z8WxT+Mp9KXxnc+H0swsX7m1MiSORuYswYYxkL07Gu21bwzq2o6pLdWvi3UbCJwuLeFEKpgYOMjvjP41Q1O50HXb/UNA1vw7eBkyI5nsiwn4+9HIoOD+X9KAOj0W3kttDs4n1E6kwjBN2xyZs87vpzx7ValJWWIsQACcn8K4XS9R1HwFonhrQrzSbq8SciO4u4zlLUvJhVPBzjeB1HTjNa0fhXVI9REtx4t1G6gk8wG2dECEMpGMgZ4z+lAGNonjW28U61NNc6wunaatwbewtlfy3vGHV2frg5GFBHvmt2+1qbQfF2n2FxKZLHWd8cW85ME64OAf7rA9OxHoa82HhvVf+FdTeC30+f+14tWDQ/u22Mh/wCWofGNuN3OeK67x1ZzXXiLwTp8TtJcJf8AnM3fbGFLt+lAHf8AzbB0pkm7zYun3j/I08jK9TTJB+9h5P3j/I0AP+bd26UDdz060uPm+8elIB1+Y9aAE+by+3Smy7t8XT7/APQ07H7vqelJKPni5P3/AOhoAd827t0oG7J6daMfN949KUDk/MetADfm8vt0psu7dF0+/wD0NPx+76npTJR80XJ+/wD0NAD/AJt3bpSjdk9OtJj5vvHpSgcn5j1oAT5tnbpWbrOfO0vp/wAfy/8AoD1pY+TqelZ2sj99pfJ/4/l/9AegDR+bd26UDdk9OtLj5vvHpQByfmPWgBPm2Hp3pk27930++Kfj5Dye9MmH+r5P3xQBJ824dOlIN2T0pcfMPmPSgDk/MaAE+bYenemTbtqdPvr/ADp+PkPJ70yYfLHyfvr/ADoAk+bcOnSkG7celLj5h8x6UgHzH5jQAfNtPTvTJt2xOn31/mKfj5T8x70yYfInJ++v8xQA87tw6UDduPSlI+YfMaQD5j8xoABu2np3pku7y16feX+Yp4Hyn5j3pko/dryfvL/MUAPO7cOlA3bj06UEfMPmNKB8x+Y9KAEG7B6dajm3eSOn3l/mKkA4PzHrTJh+5HJ+8v8AMUAPO7cOlL8249OlIR8w+Y0uPmPzHpQAg3c9OtFKB1+Y9aKAIfPt9o+de1UtWtNP1aBbO7w0THcCpwyMOVZT2YHkGtLJ2jg9qZIT58XB7/yoAjhkiSNVkuBK4UAuQAW98Dini4gwR5g61Lk7j8p6UAnB4PWgClPDY3LwSTMH8hg6IT8obsxHcjt6VLJNB5sXzrwTn8qnydg4PamSH99Dwep/lQAn2iDJHmDGKow2FgmrS6pJL5106eUjv0ijznao7AnknqfwGNPJ3H5T0oBPPB60AQ+db7B8602SaDzIvnXhjn8jU+TsHBpkhPmxcH7x/kaAE8+33ffXpQJ7fn5161Lk7uh6UAnng9aAIfOt9n316U2SaDfF86/e/oanyfL6HpTZSd8XB+//AENADfPt93316UCe35+detS5O7oelAJyeD1oAh8632ffXpTZZoC0Xzr9/wDoanyfL6HpTJSd0XB+/wD0NACefb7vvr0oE9vk/OvWpcnd0PSgE5PynrQBk6jrdva6e9xZ+RfMhAaNLgA88ADg8k4AHvTdWniZ9KJKq321CwznB2PV5rG1ZF/0ZV2Osg2AL8ynIJx1war6yf32l8H/AI/l/wDQHoAu+fb7vvr0oE9vk/OtS5O7oelAJyeD1oAh8632H51pk00B8vDr98VYydh4PemTE/u+D98UAJ59vuHzr0oE9vk/OtS5O4fKelAJyeDQBD59vsPzr3pk00BVMOv3xVjJ2Hg96ZMTtTg/fX+dACefb7h869KBPb5PzrUuTuHynpQCdx4NAEPn2+0/OvemSzQFFw6/fX+dWMnaeD3pkxOxOD99f5igBDPb7h860Ce33H51qUk7hwaATuPBoAiE9vg/OvemSzQeWuHX7y/zFTgnaeD3pkxPlrwfvL/MUAIZ7fcPnWjz7fcfnWpSTuHBoBO48HpQBEJ7fB+detRyzQGEAOvVf5irAJweD1pkxPkjg/eX+YoAQz2+R860efb7j869KlJORwaMncflPSgCIT2/Pzr1oqUE88HrRQAfwj8KZJ/r4vx/lRRQBJ/EfpSDofrRRQAfwD8KZJ/rofqf5UUUASfxH6Ug7/WiigA/5Zj6CmSf62H/AHj/ACNFFAEn8X4Ui9/rRRQAn/LP8KbL/rIf9/8AoaKKAH/x/hSjqfrRRQA3/ln+FNl+9F/v/wBDRRQA/wDj/ClHU/WiigBP+WZ+lZus/wCu0v8A6/l/9AeiigDT/i/CgdT9aKKAE/gP40yf/ln/AL4oooAk/iH0oHU0UUAJ/Afxpk/3Y/8AfX+dFFAEn8Q+lIPvGiigA/hP40yb7if76/zFFFAEh+8KQfeNFFAAPun8aZN/ql/3l/mKKKAHn7wpR94/SiigBB0P1NRzf6gf7y/zFFFAEh+8KX+I/SiigAXv9aKKKAP/2Q==)
The area of the quadrilateral is
{(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
=
{(4–(–3))(2–(–6)) –(–1–4)(1–(–5))}
=
{56+30} = 43 Sq. units
Question 15.Find the area of the quadrilateral whose vertices are
(–4, 5), (0, 7), (5, – 5) and (–4, – 2)
Answer:We must take all the vertices in counter clock wise direction otherwise it will give solution in negative.
So, from the figure we assume that
A (x1, y1) = (5, –5)
B (x2, y2) = (0, 7)
C (x3, y3) = (–4, 5)
D (x4, y4) = (–4, –2)
![](data:image/jpeg;base64,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)
The area of the quadrilateral is
{(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
=
{(5–(–4))(7–(–2)) –(–5–5)(0–(–4))}
=
{81+40} = 60.5 Sq. units
Question 16.If the three points (h, 0), (a, b) and (0, k) lie on a straight line, then using the area of the triangle formula, show that
where h, k ≠ 0.
Answer:Let A(h,0) B(am) and C (0, k) are the three points
Since the three points A(h,0) B (a, b) and C (0, k) lie on a straight line we can say that the three points are collinear.
So, area of triangle ABC = 0
Area of triangle = ![](data:image/png;base64,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)
x1 = h, x2 = a and x3 = 0
y1 = 0, y2 = b and y3 = k
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARoAAAAgCAMAAAD6+0f3AAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZYOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZpDbZrb/kDoAkGYAkGY6kLbbkNvbkNv/tmYAtmY6tpA6tpBmttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///b+9nWpAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADg0lEQVRoQ+1Y2WLTMBC00gDhCPQIAWpampaCSKF2ov//N6RdHbuSbMuleQnWi1tb0s6MZldSqmpqkwKHUEDdvjrEtEcw568P60marnWUkzSTNKOTfHJNp2REmoe1uDD91Pflz9ESFw+Qp/f9fQ8S3kWVYn5dCFVtXj7aro04hz9VrZ9SzG7wvRQnJUKFAUOR27eXfV0g/L81j4WAclHVncDw6kqIZc8qSaEbWEUr8sIqA/83TppKFklDBuR5BSjta6t6rqOqLZx+5P3EPPjAonJRLU/z2NVl1PYr3Ksa7N4jjapzy05QWMqsH4UicRGyzYZPkPOgKTH2PSdN5aJugCB0aayDBkyqapDGPp5dGgqlxzYsPEXOpUmJDUvjom5gYUAgZ4eh/MUh7QIt0cy+3WKuyZMfV2L2JQznbti+EeLMLoPOz3lIX9aPQnFJk0HkwifIuTQpsUQawBJYmEXHTQYsgGm1X/X4l6DDmuK82Ih392pjXCdnek+RxHoUhQ70iN3MQPNfXkIOZdN5zrThU+RJdkbEEmkAS2BhjAJR9yujEAZw9VV/M0UXmq+xlAiMDNJo94CjpelMrZfUGlhqM7BZ0iLCJaRQ0KC5xqQJyA0LUuBiYsCVfPdYgIDLSxsVHpkZOBwrlHnZLnAXd/sRYATO+MbNm1PXSbMNysSrwKF4aZLFsuGj7nHQiFgyiwaPWAKLQEF9BX9jQhVdk2wB9K7JS4PZShfw4dNC4NoIwc2QusZBKXRNp1UzxLhrHJacNKwM40lhIKGsO0ZKAxntXNMu2CLkyjBCGaw1bgNxyKP1SD8nCYVYuDSk1mDxLNy8bcqEHconFNaaPMoQG/KbnROYNBQKnSwqOC58gpwXuJRYWmuAu/e+p2Czw2xOtJZ11D54HZ9rvKpS6E3ojhRuiqJdXFS7z1CGzQDYq1xjaCmUgnNNgpxLkxKLEspioWWYn2uq3VqI07IrCT8N68oxu2mhishzfWafd1HWh5+zPytxCQNU7a4dSU2iUApOwwnyaFtMiNHvHsv7wEJvJ7YS4ml4TIvuUGOGjuxbdIcaOedw9+gONTjgt3aVOcqaxm7egyOf3uHwN+8ctuTmPUBg//G62vryMP1eE+8JfT8RPN0bxzBShl35GOg8I4etKzXPOOdxTCUnZToWstHKNL0/1R6HA8az2K/MnXaSZrxy04j/UIG/PHaIx8vtC6UAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ 0 = hb + ak – kh
⇒ hab + ak = kh
Divided by (kh) on both sides we get
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFsAAAAhCAMAAACBUScMAAAAAXNSR0IArs4c6QAAAG9QTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6Ojo6OpDbZgAAZma2ZrbbZrb/kDoAkGYAkGY6kLb/kNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm29v/2////7Zm/9uQ/9u2//+2///br4xYJwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABfElEQVRIS+2WDVPCMAyGGzas3yDSKVRlyP7/byRdk027pq23U887ezfKkTfvQtb2mVJ/dFjQfeU8p/5GTjPxMt5b8Zwyz2lCL/sFb9ZK9+f4MF80sHjEut28ErLeNwA32DitDIAkGuOWPG216x6wdlPtlK32UfPT7b1qlgf07l7u3uSucZw9Xf3ucr06Xm3FxNZ5A+Bnwpvi7Fnk/XrdZ1l9vKTHE70Bx9mzQbG7zPLQPQk9aRdbZeu907Wg5co5Tp4tPiR3KYOl1UJLTmuon0F7HW2ISOEcJ0+5d91aXg9zz4l/72kHsz3BHUoDV1F0DAL64s+tIC0QSY+yUKZUtu4ZiyXrne+JePes93fW/bveRbwsEo3MHbhaxMsi0chcloeMEw5m/3MGmBNeEgOZcZJ3D8qkKMJLYiAzTvD2oEyKnPdnGTOQORT3DkAorL/QK2RcEQglb+LpB172DGRuRtMCEErbJsJLx8CBm7G8AIT4FhQdxbycsaV/IPUMbwgfbPsFvOYAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
Hence proved.
Question 17.Find the area of the triangle formed by joining the midpoints of the sides of a triangle whose vertices are (0, – 1), (2,1) and (0,3). Find the ratio of this area to the area of the given triangle.
Answer:Let A (0, –1), B (2, 1) and C (0, 3) are the vertices of the triangle.
D, E and F are the mid–points of the sides AB, BC and AC respectively.
![](data:image/jpeg;base64,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)
Mid – point formula = ![](data:image/png;base64,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)
Mid – point of AB![](data:image/png;base64,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)
![](data:image/png;base64,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)
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D = (1,0)
Mid – point of BC = ![](data:image/png;base64,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)
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Mid – point of AC = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Area of Δ ABC:
Area of triangle = ![](data:image/png;base64,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)
x1 = 0, x2 = 2 and x3 = 0
y1 = –1, y2 = 1 and y3 = 3
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 4 sq. units
Area of Δ DEF:
Area of triangle = ![](data:image/png;base64,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)
x1 = 1, x2 = 1 and x3 = 0
y1 = 0, y2 = 2 and y3 = 1
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAN0AAAAgCAMAAAChFo3vAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6Ojo6OjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkGYAkGY6kLbbkNv/tmYAtmY6tpA6tpBmttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///brpB+jgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACkElEQVRYR+1Ya3faMAy1+1hL17Etpe0yWLpR95UQ//+fV0eyGtlxbChwcsjwF+BI8dWVrmQHIY7rgDOgHy4POPpE6E/fZiNmJ4Q6sjtY8f4vtXuZyZ9NkfS/68edFUtNl9G99gKm5PkCUHVx8YbwpfwB33SOnzta1ddfkZ32BKb/ygZVSbOwYvkXSw5/mkrK8z+RwF5uY1Y9l/IaylZd9fvpHMBE6x0ETJjNMxQMgREb2q7OcHiWpyDLQt6LVYbfcemcF2E1k9zom5vdVzl6KExbaFkw7h0AS5iFYMEQWOFHB+x0jiSL5kNBfUPsqulSxdiVJ6ZgJT7eXzwC494Bdgmz4MEQWOGlFH9XE0bIhheqnaEeYweps3Ig+XWLR2DcOwCWMIM+KBgLRnn7wEQHyBMtp7yuMuPsUPZ1hgkEHQSFiWCuNzhysIQZ/NtUI1idYUPTsmw5O6eOXt+tw45a21fJB6YFQz9nEHTZ9Zp9djaj5qOZmGYZh2qCJwPTG1NUYR2lbEvLPDtmN15i13GzW3jsfLeEOcxO/z6J1s6X7meUaSdU39B0akdDu1eZTN9+LFyZbe04Pb/vikt3kG/CTthBgOJP9Z3r3WGXMq/Td8KbmXBwlKw3N2IHZ4kduX6HtymlxubeaHXAEmaHnQXrCs857+Dc0EWMHZ+u/mnejEuaA+nzjnsH2CXMwI6C6TnvOncVGCR97PR8Yqxnd3zmOvdJc32QU5R2+q7SXDbIO8AuZebB9N5VnHsm78jtvq9zz9wOgT3t3jNfTdK+o3V87wj1zUI8k2RH+H4Xf0PZmWIG20i5N7LB4tgL8LNtu71sPvSmaszkSkOujP3xMXT2t8Gvs+bIHiu7bTJzfHbQDLwDg9pcNxvI+3IAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 1 sq. units
Area of triangle ABC: Area of triangle DEF
4: 1
Find the area of the triangle formed by the points
(0, 0), (3, 0) and (0, 2)
Answer:
(0, 0), (3, 0) and (0, 2)
Area of triangle =
x1 = 0, x2 = 3 and x3 = 0
y1 = 0, y2 = 0 and y3 = 2
= 3sq. units
Question 2.
Find the area of the triangle formed by the points
(5, 2), (3, –5) and (–5, –1)
Answer:
(5, 2), (3, –5) and (–5, –1)
Let A (–5, –1) B (3, –5) and C (5, 2) be the vertices of triangle.
Area of triangle =
x1 = –5, x2 = 3 and x3 = 5
y1 = –1, y2 = –5 and y3 = 2
= 32 sq. units
Question 3.
Find the area of the triangle formed by the points
(–4, –5), (4, 5) and (–1, –6)
Answer:
(–4, –5), (4, 5) and (–1, –6)
Let A (–4, –5) B (–1, –6) and C (4,5) be the vertices of triangle.
Area of triangle =
x1 = –4, x2 = –1 and x3 = 4
y1 = –5, y2 = –6 and y3 = 5
= 19 sq. units
Question 4.
Vertices of the triangles taken in order and their areas are given below. In each of the following find the value of a.
Vertices: (0, 0), (4, a), (6, 4)
Area (in sq. units): 17
Answer:
Vertices of triangle A (0, 0), B (4, a) and C (6, 4)
Area of triangle = 17 sq. units
Area of triangle =
x1 = 0, x2 = 4 and x3 = 6
y1 = 0, y2 = a and y3 = 4
⇒
⇒
⇒
⇒ 17 × 2 = 16 – 6a
⇒ 34 = 16 – 6a
⇒ 34 + 6a = 16
⇒ 6a = 16 – 34
⇒ 6a = – 18
⇒
⇒ a = –3
Therefore, the required vertices are (4, –3)
Question 5.
Vertices of the triangles taken in order and their areas are given below. In each of the following find the value of a.
Vertices: (a, a), (4, 5), (6, –1)
Area (in sq. units): 9
Answer:
Vertices of triangle A (a, a), B (4, 5) and C (6, –1)
Area of triangle = 9 sq. units
Area of triangle =
x1 = a, x2 = 4 and x3 = 6
y1 = a, y2 = 5 and y3 = –1
⇒
⇒
⇒
⇒
⇒ 9 × 2 = 8a – 34
⇒ 18 = 8a – 34
⇒ 8a = 18 +34
⇒ 8a = 52
⇒
⇒
Therefore, the required vertices are
Question 6.
Vertices of the triangles taken in order and their areas are given below. In each of the following find the value of a.
Vertices: (a, –3), (3, a), (–1,5)
Area (in sq. units): 12
Answer:
Vertices of triangle A (a, –3), B (3, a) and C (–1, 5)
Area of triangle = 12 sq. units
Area of triangle =
x1 = a, x2 = 3 and x3 = –1
y1 = –3, y2 = a and y3 = 5
⇒
⇒
⇒
⇒ 12 × 2 = a2 – 4a + 27
⇒24 = a2 – 4a + 27
⇒ a2 – 4a + 27 – 24 = 0
⇒ a2 – 4a + 3
⇒ a2 – 3a – a + 3 = 0
⇒ a (a – 3) – (a – 3) = 0
⇒ (a – 3) (a – 1) = 0
a – 3 = 0 or a – 1 = 0
a = 3 or a = 1
Therefore, the required vertices are (3, –3) or (1, –3)
Question 7.
Determine if the following set of points are collinear or not.
(4, 3), (1, 2) and (–2, 1)
Answer:
(4, 3), (1, 2) and (–2, 1)
Let A (4, 3) B (1, 2) and C (–2, 1) be the vertices of triangle.
Area of triangle =
x1 = 4, x2 = 1 and x3 = –2
y1 = 3, y2 = 2 and y3 = 1
⇒
⇒
⇒
⇒
Therefore, the given points are collinear.
Question 8.
Determine if the following set of points are collinear or not.
(–2, –2), (–6, –2) and (–2, 2)
Answer:
(–2, –2), (–6, –2) and (–2, 2)
Let A (–2, –2) B (–6, –2) and C (–2, 2) be the vertices of triangle.
Area of triangle =
x1 = –2, x2 = –6 and x3 = –2
y1 = –2, y2 = –2 and y3 = 2
⇒
⇒
⇒
⇒
Therefore, the given points are non – collinear.
Question 9.
Determine if the following set of points are collinear or not., (6, –2) and (–3, 4)
Answer:
Let A B (6, –2) and C (3, –4) be the vertices of triangle.
Area of triangle =
, x2 = 6 and x3 = 3
y1 = 3, y2 = –2 and y3 = 4
⇒
⇒
⇒
⇒
Therefore, the given points are collinear
Question 10.
In each of the following, find the value of k for which the given points are collinear.
(k, –1), (2, 1) and (4, 5)
Answer:
(k, –1), (2, 1) and (4, 5)
Area of triangle =
x1 = k, x2 = 2 and x3 = 4
y1 = –1, y2 = 1 and y3 = 5
if points are collinear then, area of triangles are collinear.
⇒
⇒
⇒
⇒ k + 6 – 2 – 5k = 0
⇒4 – 4 k = 0
⇒ 4 = 4k
⇒
⇒ k = 1
Question 11.
In each of the following, find the value of k for which the given points are collinear.
(2, – 5), (3, – 4) and (9, k)
Answer:
(2, –5), (3, 4) and (9, k)
Area of triangle =
x1 = 2, x2 = 3 and x3 = 9
y1 = –5, y2 = –4 and y3 = k
if points are collinear then, area of triangles are collinear.
⇒
⇒
⇒
⇒
⇒ k – 2 = 0
⇒ k = 2
Question 12.
In each of the following, find the value of k for which the given points are collinear.
(k, k), (2, 3) and (4, – 1)
Answer:
(k, k) (2, 3) and (4, –1)
Area of triangle =
x1 = k, x2 = 2 and x3 = 4
y1 = k, y2 = 3 and y3 = –1
⇒
⇒
⇒
⇒ 7k – 2 – k – 12 = 0
⇒ 6k – 14 = 0
⇒ 6k = 14
⇒
Question 13.
Find the area of the quadrilateral whose vertices are
(6, 9), (7, 4), (4,2) and (3,7)
Answer:
When the vertices of a quadrilateral is given then its area is given by {(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
We must take all the vertices in counter clock wise direction otherwise it will give solution in negative.
So, from the figure we assume that
A (x1, y1) = (7,4)
B (x2, y2) = (6, 9)
C (x3, y3) = (3, 7)
D (x4, y4) = (4, 2)
The area of the quadrilateral is {(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
= {(7–3)(9–2) – (6–4)(4–7)}
= {28–(–6)} = 17 Sq. units
Question 14.
Find the area of the quadrilateral whose vertices are
(–3, 4), (–5, – 6), (4, – 1) and (1, 2)
Answer:
When the vertices of a quadrilateral is given then its area is given by {(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
We must take all the vertices in counter clock wise direction otherwise it will give solution in negative.
So, from the figure we assume that
A (x1, y1) = (4, –1)
B (x2, y2) = (1, 2)
C (x3, y3) = (–3, 4)
D (x4, y4) = (–5, –6)
The area of the quadrilateral is {(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
= {(4–(–3))(2–(–6)) –(–1–4)(1–(–5))}
= {56+30} = 43 Sq. units
Question 15.
Find the area of the quadrilateral whose vertices are
(–4, 5), (0, 7), (5, – 5) and (–4, – 2)
Answer:
We must take all the vertices in counter clock wise direction otherwise it will give solution in negative.
So, from the figure we assume that
A (x1, y1) = (5, –5)
B (x2, y2) = (0, 7)
C (x3, y3) = (–4, 5)
D (x4, y4) = (–4, –2)
The area of the quadrilateral is {(x1–x3)(y2–y4) – (x2–x4)(y1–y3)}
= {(5–(–4))(7–(–2)) –(–5–5)(0–(–4))}
= {81+40} = 60.5 Sq. units
Question 16.
If the three points (h, 0), (a, b) and (0, k) lie on a straight line, then using the area of the triangle formula, show that where h, k ≠ 0.
Answer:
Let A(h,0) B(am) and C (0, k) are the three points
Since the three points A(h,0) B (a, b) and C (0, k) lie on a straight line we can say that the three points are collinear.
So, area of triangle ABC = 0
Area of triangle =
x1 = h, x2 = a and x3 = 0
y1 = 0, y2 = b and y3 = k
⇒
⇒
⇒
⇒ 0 = hb + ak – kh
⇒ hab + ak = kh
Divided by (kh) on both sides we get
⇒
⇒
Hence proved.
Question 17.
Find the area of the triangle formed by joining the midpoints of the sides of a triangle whose vertices are (0, – 1), (2,1) and (0,3). Find the ratio of this area to the area of the given triangle.
Answer:
Let A (0, –1), B (2, 1) and C (0, 3) are the vertices of the triangle.
D, E and F are the mid–points of the sides AB, BC and AC respectively.
Mid – point formula =
Mid – point of AB
D = (1,0)
Mid – point of BC =
Mid – point of AC =
Area of Δ ABC:
Area of triangle =
x1 = 0, x2 = 2 and x3 = 0
y1 = –1, y2 = 1 and y3 = 3
⇒
⇒
⇒
⇒
⇒ 4 sq. units
Area of Δ DEF:
Area of triangle =
x1 = 1, x2 = 1 and x3 = 0
y1 = 0, y2 = 2 and y3 = 1
⇒
⇒
⇒
⇒
⇒ 1 sq. units
Area of triangle ABC: Area of triangle DEF
4: 1
Exercise 5.3
Question 1.Find the angle of inclination of the straight line whose slope is
(i) 1 (ii) √3 (iii) 0
Answer:If θ is the angle of inclination of the line, then the slope of the line is
m = tan θ where 0° ≤ θ ≤ 180°, θ ≠ 90°
i) tan θ = 1
⇒ θ = 45°
ii) tan θ = √3
⇒ θ = 60°
iii) tan θ = 0
⇒ θ = 0°
Question 2.Find the slope of the straight line whose angle of inclination is
(i) 30° (ii) 60° (iii) 90°
Answer:If θ is the angle of inclination of the line, then the slope of the line is m = tan θ
i) Given that m = tan 30°
⇒ ![](data:image/png;base64,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)
ii) Given that m = tan 60°
⇒ m = √3
iii) Given that m = tan 30°
⇒ ![](data:image/png;base64,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)
∴ The slope is undefined.
Question 3.Find the slope of the straight line passing through the points
(i) (3, –2) and (7, 2) (ii) (2, –4) and origin
(iii)
and ![](data:image/png;base64,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)
Answer:Slope of straight line passing through the points (x1, y1) and (x2, y2) is given by
![](data:image/png;base64,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)
i) Slope of straight line passing through the points (3, –2) and (7, 2) is
![](data:image/png;base64,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)
ii) Slope of straight line passing through the points (2, –4) and (0, 0) is
![](data:image/png;base64,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)
iii) Slope of straight line passing through the points (1 + √3, 2) and (3 + √3, 4) is
![](data:image/png;base64,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)
Question 4.Find the angle of inclination of the line passing through the points
(i) (1, 2) and (2, 3) (ii)
and (0, 0)
(iii) (a, b) and (–a, –b)
Answer:Slope of straight line passing through the points (x1, y1) and (x2, y2) is given by
![](data:image/png;base64,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)
i) Slope of straight line passing through the points (1, 2) and (2, 3) is
![](data:image/png;base64,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)
m = 1
If θ is the angle of inclination of the line, then the slope of the line is
m = tan θ where 0° ≤ θ ≤ 180°, θ ≠ 90°
∴ tan θ = 1 ⇒ θ = 45°
ii) Slope of straight line passing through the points (3, √3) and (0, 0) is
![](data:image/png;base64,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)
m = √3
If θ is the angle of inclination of the line, then the slope of the line is
m = tan θ where 0° ≤ θ ≤ 180°, θ ≠ 90°
∴ tan θ = √3 ⇒ θ = 30°
iii) Slope of straight line passing through the points (a, b) and (–a, –b) is
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAArCAMAAAAaGrsZAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOjpmOmaQOma2OpC2OpDbZgAAZjo6ZmY6ZpDbZrb/kDoAkGY6kLbbkNv/tmYAtmY6tpA6ttv/tv/btv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///b5U7UBAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA8ElEQVRIS91U2RKCMAxs8Kh4oCKCVz2g2v//Q9OC4Ex1aDqOI+ah04dsmuxuytiPh4Dewa9F0SXgPoFgSZ9TBOMdExCTkSJIGbtFQzrQsJoNciqylOOrwHLGGbVTJHSYq61GE0NME4A+HUd8puvpBUB85DBh5qCE5GGK8oRrZgzsHpLjjkiOXjC3OjJ4xJt6FdCgW9esLgZVeoMmt2q9+NlWn/sp9AdSkUPZFslRyOZwH/F/M9VpBERPGzLUCr+cjObphkUHa76m3AuoznMtPFlEPaPLFlmFC82LT6va09dF+97ao2xQxUvkMSSZFQtwByy0D9N+hg9bAAAAAElFTkSuQmCC)
If θ is the angle of inclination of the line, then the slope of the line is
m = tan θ where 0° ≤ θ ≤ 180°, θ ≠ 90°
∴ ![](data:image/png;base64,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)
Question 5.Find the slope of the line which passes through the origin and the midpoint of the line segment joining the points (0, – 4) and (8, 0).
Answer:We need to find slope of a line passing through origin (0, 0) & mid–point of P (0, –4) and Q (8, 0)
Let M mid–point of P (0, –4) and Q (8, 0)
Mid–point formula M (x, y) ![](data:image/png;base64,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)
Mid–point of PQ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
We know that,
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of line between points O (0, 0) and M (4, –2)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAAAqCAMAAACdrGVEAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZmZmZrbbZrb/kDoAkGY6kLbbkNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///b2YM4KQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABdUlEQVRYR+1Wf1PDIAwFfww33aauWrRz0PH9v6NJoO1Zm24FevO85Q+ud9C85CWEJ8TVPAPu61nKu5Klw71JufpIZUvLV1Fvbj8ZP664P9QFu30uul7AyUrumPPmBtI07Pa5KHSOd6MxjeMGQ0k28jVkSBii4JpqVnGEeRS/TjMtG0POsc2KdeOhvxeP0o/JFTzrDWPpddGLETpC9dtkp9HWna6Qc8O5oR5P72T7AMVxmkPB9oopfi/nUG+WkhoG0OPkDosl9vrfH2XASLnbK/kkaJnNrFqW8EYs30UVRtMcUDRTrYKe/zldf83FJPCAQlDcDG8B2ok87SP47qCSIuZ/5nK5IGPRmdJkDtVPfgm4KKyCC9Mt0cHO9ePY9cojKiFyq/hLnElUosx64VGyiUq9JlfDlktUmtWBR8klKo/bUpxCSdYVJCgBpfKDbyZRCY+St+Hxmk9UUi4nqp9jlIygZBKVmMTI3c8kKgEEasNT9o9F5TdXgxutSvV1cgAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
Hence, slope of line is ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6AGaQAGa2OgA6OmZmOma2OpDbZgAAZgA6ZjoAZrb/kLbbkNv/tmYAtmY625A625Bm27Zm29u22////7Zm/9uQ//+2///bgWo6oAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbElEQVQ4T2NgoAKQFGDBaooQGzd2CQYGQdpLSPIzi2JzliAjEHBQwdsUGcEPcgUYMPEBDYKyKTKSnppFuBkZ2bFYKMHFyyAM9hEWIM6KQ0IQR3QIY7MCaLAgDnExoLgYD6bNEpyg8MUiQTDIAKWvApSMomtfAAAAAElFTkSuQmCC)
Question 6.The side AB of a square ABCD is parallel to x–axis. Find the
(i) slope of AB (ii) slope of BC (iii) slope of the diagonal AC
Answer:i). Slope of AB
Since the side AB is parallel to x–axis,
Slope of side AB = 0
Therefore, the slope of AB is 0
ii). Slope of BC
The angle formed by the side BC is 90°
m = tan θ
θ = 90°
m = tan 90°
![](data:image/png;base64,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)
Therefore, the slope of BC is not defined.
iii). Slope of the diagonal AC
The diagonal AC is the angle bisector of the angle ∠BAD
∴ θ = 45°
m = tan θ
= tan 45°
= 1
Therefore, the slope of the diagonal AC is 1
Question 7.The side BC of an equilateral 3ABC is parallel to x–axis. Find the slope of AB and the slope of BC.
Answer:Slope of line BC:
The side BC is parallel to x–axis.
Slope of the side BC = 0.
Slope of line AB:
Since, this is an equilateral triangle each angle is 60°
∴ θ = 60°
m = tan θ
= tan 60°
= √3
Hence, the slope of AB is √3 and the slope of BC is 0
Question 8.Using the concept of slope, show that each of the following set of points are collinear.
(2, 3), (3, –1) and (4, –5)
Answer:(2, 3), (3, –1) and (4, –5)
Let A (2, 3), B (3, –1) and C (4, –5) be the given points
If the given point is collinear then,
Slope of AB = slope of BC
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of AB:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAAAsCAMAAABL9YZZAAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOmaQOma2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bpL+C2gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABa0lEQVRYR+1Xf3PCIAwl7oc463Rbq+K6dYXx/b/igKHVrRAKVM878wfXO0pe8kKTV0JudmYGmpetE1F+PgE8uPdDQxUruHt3vswmGyKqSSoML3a1D2WqAqihDI3Z/Z4PxZxinjCC4REUsS52wb4ic/leAhRfw1EY7M0W9SiXf3vafUPPwBghLTwPT+bvCbT6nI6LIivtv81yk90fndQfpKgeI8p/wphcU3UZ7l8dxAu9vUgFSa/qzcNlGWgByg8KC2KW0YzT+VYNgfmG1Mnzxh0kp2rKmDZgng7W2/uic7UoBgqdaoeOPOzB+u6goqP1H3TlckHGojM1ndlWP8Mk6I+Dq7ZZdkt0sGMcRFSjbFY5RCWiGhm8EbFMlhXMrxqR7UHsIqoxy8BHVWMG6YqrxoB+FcJcpxr72sSvTMpgPtUoK30/cphPNbJpukBCVWOtpV6byBmmGvlM/eVJlohCENVor0MqSo6yXqmPHwgXF/Xtnny2AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
= –4
Slope of BC:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= –4
Slope of AB = Slope of BC = –4
Therefore, the given points are collinear.
Question 9.Using the concept of slope, show that each of the following set of points are collinear.
(4, 1), (–2, –3) and (–5, –5)
Answer:(4, 1), (–2, –3) and (–5, –5)
Let A (4, 1), B (–2, –3) and C (–5, –5) be the given points
If the given point is collinear then,
Slope of AB = slope of BC
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of AB:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Slope of BC:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the given points are collinear.
Question 10.Using the concept of slope, show that each of the following set of points are collinear.
(4, 4), (–2, 6) and (1, 5)
Answer:(4, 4), (–2, 6) and (1, 5)
Let A (4, 4), B (–2, 6) and C (1, 5) be the given points
If the given point is collinear then,
Slope of AB = slope of BC
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of AB:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Slope of BC:
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore, the given points are collinear.
Question 11.If the points (a, 1), (1, 2) and (0, b + 1) are collinear, then show that
.
Answer:Let A (a, 1), B (1, 2) and C (0, b + 1) be the given points.
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of AB ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Slope of BC ![](data:image/png;base64,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)
![](data:image/png;base64,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)
If three points are collinear, then slope of AB is equal to slope of AC
∴ slope of AB = slope of BC
![](data:image/png;base64,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)
⇒ 1 = (–b + 1)(1 – a)
⇒ 1 = –b(1 – a) + 1(1 – a)
⇒ 1 = –b + ab + 1 – a
⇒ –b + ab – a = 0
⇒ ab – b = a
Dividing both sides by ab, we get
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence proved.
Question 12.The line joining the points A (–2, 3) and B (a, 5) is parallel to the line joining the points C (0, 5) and D (–2, 1). Find the value of a.
Answer:Line joining points A (–2, 3) and B (a, 5) IA parallel to the line joining the points C (0, 5) and D (–2, 1)
The two lines are parallel only when their slopes are equal.
∴ slope of AB = slope of CD
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of AB is:
A (–2, 3) and B (a, 5)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Slope of CD is:
C (0, 5) and D (–2, 1)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAAArCAMAAABW8LbhAAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOjqQOmaQOma2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bl2g7pgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABc0lEQVRYR+1XbVPCMAxufKPOKaCbSHHgOvv//6JNmOwYS92WTu+E50OPu2Z5fZoEpS7ok4Htuo9UWGa35JVYDR5XYivVAq7fWTeszuRhKGXTTTG9Fe/p+VhJNNyKi3+cMYOMItS0cq8vqsrlHAvXhQhm9ZOcaKHqk/bP+WxaKwajiBNLIO3G31V54EH1C9KtsIfcPDPS1dLfppt+ui5S/z0DJUC21fCo6JgMVidrVUDypooYjYfxk+YavdPjCXfSAkVx1lbI1I9z9NB8h/2odTemRB7zH3Ox/GHGRkdaQlP9CGOn2w9c0LLmGO3shB+63QJCW4RbATyIB4ABv0XM2Wnl8ruPCMPM4GQvsLCdKLGZUN3l4NUYjDLKkqEU6eoCJgyt4CkF35f2VvbnMJyukfnhkbXvxltp++RyfrX7zph8+TOzQDrq6os7SoE5Lzk1xHE5k+29fxCO9tUuIL3GFL+lq643mxL/7xPSwQwbxsfflv4C20gaG+j114sAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAXCAMAAABd273TAAAAAXNSR0IArs4c6QAAAEtQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OpDbZgAAZjoAZrbbZrb/kDoAkNv/tmYAtv//25A627aQ2////7Zm/9uQ/9u2///bQDTJKwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQoU9WSSw6AMAhEwU/9t1ht5f4nFY1uTMB1ZzsT5kEAKFi8TYhN0DcgnOEY61VNkBMr4mIfIf0F6K4gfFV9mHJnN7AfTAL2F6dRQW43B8RW/KSX5F6ImfTAs5uNWc4rnWGQAuTIKGdtAAAAAElFTkSuQmCC)
As per the property, the two lines are parallel only when their slopes are equal.
i.e. Slope of AB = Slope of CD
⇒ ![](data:image/png;base64,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)
⇒ 2 = 2(a + 2)
⇒ 2 = 2a + 4
⇒ 2a = 2 – 4
⇒ 2a = –2
⇒ ![](data:image/png;base64,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)
⇒ a = –1
Question 13.The line joining the points A (0, 5) and B (4, 2) is perpendicular to the line joining the points C (–1, –2) and D (5, b). Find the value of b.
Answer:Line joining points A (0, 5) and B (4, 2) IA parallel to the line joining the points C (–1, –2) and D (5, b)
The two lines are perpendicular only if the multiplication of their slope is equal to 1.
∴ (Slope of AB) × (Slope of CD) = 1
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of AB is:
A (0, 5) and B (4, 2)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Slope of CD is:
C (–1, –2) and D (5, b)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
The two lines are perpendicular only if the multiplication of their slope is equal to 1.
∴ (Slope of AB) × (Slope of CD) = –1
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ –b – 2 = –1 × 8
⇒ –b – 2 = –8
⇒ –b = –8 + 2
⇒ – b = – 6
⇒ b = 6
Question 14.The vertices of ΔABC are A (1, 8), B (–2, 4), C (8, –5). If M and N are the midpoints of AB and AC respectively, find the slope of MN and hence verify that MN is parallel to BC.
Answer:Given: vertices of triangle ABC i.e. A (1, 8), B (–2, 4), C (8, –5)
M and N are mid – points of AB and AC.
Finding co–ordinates of M and N:
We know that,
M is the mid–point of AB
x1 = 1, x2 = –2
y1 = 8, y2 = 4
Mid–point formula M (x, y) ![](data:image/png;base64,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)
Mid–point of AB ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
N is the mid–point of AC
x1 = 1, x2 = 8
y1 = 8, y2 = –5
Mid–point of AC ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Slope of MN:
![](data:image/png;base64,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)
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Verification of MN and BC are parallel:
If MN and BC are parallel, then their slopes must be equal.
Slope of BC:
B (–2, 4) and C (8, –5)
Slope of BC ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ Slope of MN = Slope of BC = ![](data:image/png;base64,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)
Hence, MN is parallel to BC.
Question 15.A triangle has vertices at (6, 7), (2, –9) and (–4, 1). Find the slopes of its medians.
Answer:Given: Vertices of triangle A (6, 7), B (2, –9) and C (–4, 1)
To find the slopes of medians. We need to know the mid–points of AB, BC and AC.
D, E and F are mid–points of AB, BC and AC respectively.
![](data:image/jpeg;base64,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)
Mid – point formula = ![](data:image/png;base64,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)
Mid – point of AB = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Mid – point of BC = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Mid – point of AC = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEcAAAAqCAMAAADWBLapAAAAAXNSR0IArs4c6QAAAGNQTFRFAAAA///b//+22///tv///9u2/9uQkNv//7Zm27aQ27ZmkLbbZrb/Zrbb25A6ZpDbOpDbOpC2tmY6tmYAAGa2kDoAZjoAOjoAADqQADpmZgA6ZgAAOgA6OgAAAABmAAA6AAAADhSbFAAAAAF0Uk5TAEDm2GYAAAEVSURBVHja1ZXNEoIwDIQDqPwItUpREZG8/1M6BcehkA6m9IA9cFhgSbYpH8DfriC9ItaxKWY3xO7E8imxgH3z2o010RUAGUqWj9IvYj4usVL9nXvEbS4hfRQ7pNLsK0HdVxdzbQ5tbgppi8i3CSo59ZXTzH6yUVNBJzxcWTs2eT5sFBHa0hL6s4mc1cPc98MzBghKIyGB54g9h9gvOTsX9XGLp7gayuUPE1B7J1b7DMMVPvz4+AjIo0+miEnhpT9s2HJNX2+kNaIeX/k49uUn588/ZG0TOmY+NtY0MeZp2OR2xjJ4einsjHXgKak58NSuMXkKDrggeWrRuDy1aA5T4zJJM55aNDZPac2Bp5TmwlOSsVtYb0HYGuXFVeGFAAAAAElFTkSuQmCC)
F = (1, 4)
Slopes of median of triangles:
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
A (6, 7) and E (–1, –4)
Slope of AE![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAAqCAMAAADRRmi8AAAAAXNSR0IArs4c6QAAAFdQTFRFAAAA///b//+22///tv///9u2/9uQttv/kNv//7Zm27aQZrb/25A6ZpDbOpDbtmYAkGY6OmaQAGa2kDoAZjo6ZjoAADqQADo6ZgAAOgAAAABmAAA6AAAAYEMt9wAAAAF0Uk5TAEDm2GYAAACOSURBVHjaY2AYtIBLhAODRQRgFZWRYkFjEQPYBNl5IMoRLGIBQvlg08gvAwPSHIPcqVTUKM2BwSIMGAUkZWRkxHmRWaNgcAGM5EwRkCEEBo9ThzjggYSEGDPJocgNKpAlucm0lrRyAw6YJPjItJDMuGYUFiIvUXHKkBk0/KTHBaSalCQzaPjpHRecA5vvACsxEIXOPpj3AAAAAElFTkSuQmCC)
![](data:image/png;base64,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)
B (2, –9) and E (1, 4)
Slope of BF![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC4AAAAqCAMAAAD79pkTAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAA///b//+22///tv///9u2/9uQttv/kNv//7Zm27aQ27ZmkLbbZrb/25A6ZpDbOpDbtmYAkGY6Oma2OmaQAGa2kDoAZjo6ZjoAOjoAADqQADpmADo6ZgAAOgAAAABmAAA6AAAAeWO6yQAAAAF0Uk5TAEDm2GYAAACtSURBVHja7ZTLDoIwEEVPxUpBRUCs79r5/590g2KiQjEaNPGsujhp7swkF4YiXRsANd2K7E2HrDdyGgMUfoa2vt2fVHFW6ysgk3lXmFoHoLh5d+q6rGJC9dFRpIroESZ1fcJAInkffeICdWVzIAlYpDeAst6g7a59VlU6ETksQJdOZBnx560UcsGbV/+QxwwT5supKy2Ma6UF0VRaKEPqd7f+oexPKi2MptI+yhm+2RHTdrXQigAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
C (–4, 1) and D (4, –1)
Slope of CD![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAAqCAMAAAAtU0N2AAAAAXNSR0IArs4c6QAAAF1QTFRFAAAA///b//+22///tv///9u2/9uQkNv//7Zm27aQ27ZmkLbbZrb/Zrbb25A6ZpDbOpDbOpC2tmYAAGa2kDoAZjoAOjoAADqQADpmZgAAOgA6OgAAAABmAAA6AAAArsAhFAAAAAF0Uk5TAEDm2GYAAACsSURBVHja7VRdD8IgECuiwKabCNNtIvf/f6bBEZ9t4mdiH0h46NG7HgW+CKo5ikyWYATpsJkva4IRAfSyI6U5mhEYVQUmkU+owbOEyDYRNUfoRw04Qpc5W0AFz3h+g8cfr0UdtIhk+8w696t8UtWvoD2J5D3zBXMHtMT/qLEQRs0yiDhxUlRRljWJ9Ngkz2W7GkrPy/kYVnMkw71WJ6aLXg6acnDZkmn7no28Am6jDajxEWwIAAAAAElFTkSuQmCC)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAAqCAMAAAAtU0N2AAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADo6OgAAOjqQOmaQOpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrb/kDoAkGY6kNv/tmYAttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bG+F7kAAAAAF0Uk5TAEDm2GYAAAB8SURBVHja7ZM3DoBADATX5BwOODDh/9+kgha2IElMd5LnZGtt4F30maHqp1TcjhHGsLWcAeA3LjUcKvOl9kXEK/BzLY1scAEd/bM/5cmuPoRlJx190pjjnDSaSDlDg4Ez5sSAMpYqAtQxtjzd07ZKJTW70pvHG3TmettBrPGoBkTMTHpmAAAAAElFTkSuQmCC)
Question 16.The vertices of a ΔABC are A (–5, 7), B (–4, –5) and C (4, 5). Find the slopes of the altitudes of the triangle.
Answer:Let AD, BE and CF be the altitudes of a ΔABC.
![](data:image/jpeg;base64,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)
Since, the altitude AD is perpendicular to BC,
Slope of BC ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(slope of BC) × (Slope of AD) = –1 (∵ m1m2 = –1)
Let slope of AD be m1.
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGoAAAAgCAMAAADNPB2vAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOjoAOjpmOmZmOmaQOma2OpDbZgAAZjoAZjo6ZjpmZmaQZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///b0dGmIAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABVklEQVRIS+WWW1PDIBCFWW8tVmvjJaVqWqVqQ7r///e5kBDaBB1ClAflgcx0gG/PydlQxv7mkABwUiTRJpdJMBqSEkUGzpMJUzyZiShuUsjCVcESqcINZT3/XVFvt2maiVUZnL52tezv6p+q5vkjWtX1i+yjUJxr1n7RhL8EWG45tYKZ4ocHxVBc7Ig0tacqPiuYhNkzk2O+XT6UZn1k051DUcMpTq3gD+ma2r8e31bSoJql9nAUQMI6KMML7/JeAV5VJGs8qvdSvzbQyTJa3MRw075Gc94wA49KsLGwFnZQ26vsGBUYSm/YjSAXwRJcLMxnUsah+qGp7ms5yj45NZZqpzgUPtIBcPYQ6EC7LE6VowwI8kgUri5DewbXE9d0Qx3R/uciEKX/XMGIW7ScYygqQsfhFro4UORPo3wJLMGYcvgxCtwXtyyVgVTdO+/fknFF/8Ndn3X7GyMgmtR2AAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
Since, the altitude BE is perpendicular to AC,
Slope of AC ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(slope of AC) × (Slope of BE) = –1 (∵ m1m2 = –1)
Let slope of BE be m2.
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHoAAAAgCAMAAADqkpxHAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOmZmOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmaQZpDbZrbbZrb/kDoAkGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bk0qMuAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABWklEQVRYR+2W21bDIBBFoVaL9ZLipWJbxdpy+/8fdCCQLBtXMyFx8VIeyEOGbM7kMAMhl+EzcOCUPhRJhV1tyH62LcIGqL4thpaPpUTvy/xqkCuLkRWQ1VuJjNuKwvhn9PdTIR8bTq8++9Jqn+sQE5998aj3+v5D9qOduPZsW0XnKfgVXwyKXZjyBwJNnLg5AnmRKJott0TS5YbIMRUPg/bsA18cWzQYUDMoOZr9ZcWd92gYZ3cW0SfRcWWTTCcoCD9BBz7+FHQ2hFINssejO6ZAoUPCW9lBazt1GuywhJ/1abJZSvlvdHaDRagOZH+4ksWVr3PRZnV7y2qwiONhXmq5Oj0ZHGzdTP7V8Abr3uEDdP6aXxbqlZM2WM3pDL2jSRusrdb4Mzttg1W+su9wN6SJG6x3rBFN0R7rhUHrJaV3HKd60HdxwXZV6BJBTCnRTszXuORcojIy8AMydhnz2DrEEgAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
Since, the altitude CF is perpendicular to AB,
Slope of AB ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(slope of AB) × (Slope of CF) = –1 (∵ m1m2 = –1)
Let slope of CF be m3.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 17.Using the concept of slope, show that the vertices (1, 2), (–2, 2), (–4, –3) and (–1, –3) taken in order form a parallelogram.
Answer:Let A (–2, 2), B (1, 2), C (–1, –3) and D (–4, –3) be the given points taken in order.
![](data:image/jpeg;base64,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)
Now,
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of AB ![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 0
Slope of CD = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC4AAAAqCAMAAAD79pkTAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjoAOjqQOmaQOma2OpDbZgAAZjoAZrb/kDoAkGY6kLbbkNv/tmYAttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///bgMkJeQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAxElEQVRIS+1U2Q4CIRCbWQ90vUDxXJH//0spu5oYE1M2Hi/2gfBQhk4nU5EfIjrV6ZYVEO3oFOxgR/KbaiPS6IqkexS+LMYcHVpAx0mgpbcngV50Wrt0rc4JJaDs4SFvJEyhO03lw1K15nwhBf9pnQNeb8Au9MO9xOOlvNg7xJT/+r0X8TBRHdI++2otwdJz8QimvIk88p6zCK6mkzgF3+udfZ710ZSIQd6w8ZQbPBuSHi2IdJpFTCiwuZ3CzBnV2cfT7Apj1gnp6uscOgAAAABJRU5ErkJggg==)
= 0
∴ Slope of AB = Slope of CD
Hence, AB is parallel to CD. … (1)
Now,
Slope of BC ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAArCAMAAAAuY3oFAAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmaQOpDbZgAAZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bRdydOgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAsklEQVQ4T81T2RICIQyjHovHeqDVVRfk///Sgo7jgw2Mzjr2lZD0SIwZtk57ld9bkhohwBY35+0PAHNLE71HE3cbExyYIo/g7RKPcm0bFcDpL2JgkQ9ufFQZwlo2vTgMe+q/YufkrVyl07y2/fxEFcN8JlFB/DUknlcETcskpm2BYTi5saNCvPoSgIHEPRZYITocm+j01OQtcnOB2+ym8t7rIn4m0Y85gO/rYYJCuqsuegP/uQcFIzNJpAAAAABJRU5ErkJggg==)
Slope of AD = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC4AAAArCAMAAAAwqkq2AAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOmaQOpDbZgAAZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bRdydOgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAxElEQVRIS+2UyQ7CMAxEY5aGpSwBQ4Em5P//snFUIUVFZSwh1AM+9fAytSfxGDOVup3BTrylVDMcP4LCGfN2Uvja0mJkUhYrcokf8XQwwcHO9PNuNfY86wrEWXS9RdU5tR3c/Aqqh32aeXMB6T+GOFA+JuTEkHm9x/JDL/aNZvR//d2JeN/R6GqXrTCl1a7h9WDZ0oZUUdbqcIab+Rirg7uODo0N0Y4OzaTsKVcPxTU3y0S3aDt+JTGcow+pfnAURyTfMR2kvgd2LuzkcAAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
∴ Slope of BC = Slope of AD
Hence, BC is parallel to AD. … (2)
From (1) and (2), we see that opposite sides of quadrilateral are parallel.
∴ ABCD is a parallelogram.
Question 18.Show that the opposite sides of a quadrilateral with vertices A (–2, –4), B (5, –1), C (6, 4) and D (–1, 1) taken in order are parallel.
Answer:Let A (–2, 2), B (1, 2), C (–1, –3) and D (–4, –3) be the given points taken in order.
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCAElAToDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAqne3zW0sNvBAZ7ifcUTdtAC4ySew5H51YnuIbWB57iVIYoxueR2Cqo9STWZHcWWviK+0fVIXktiy+ZHiRcHGVYZHoD26Cqja+pMr20Lsd7/oTXF1C1rsyHV+cYOMgjqPSlg1C1uIJJ4pg0ced7YIxgZ70kNkRZtb3NxJcGQku5JQ8noMdB6Un9mwLYz2ke9UnVlYtIzkZGO5NV7gveKQ8QI1zBAsA3SBGZWmVWG/pgHqcckflk1qzSxwQvNKwREG5mPYVkLokk6o8svkmVIhcxhQ28xnIKtnj9ePQ1fezN5Z+RfHcd+7MTFOjZXkHPHFVNQurExc9bkNpqy3MVrI9u8IundAHIyrKTwfqFP5YrRrIh0drYWdukkkkUNw9w8kjknJ3YXk+rfofWteony390qHNb3goooqCwooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKQkAEk4A6k1lzeILYytBYRyahOOCtuMqv+8/QUAatUb7WLHT2Ec0wMzfdhjG+Rvoo5qp9i1fUeb68FlCf+WFofmP1kP9BV6x0uy05SLW3SMn7z9Wb6seTQBzHi201nxJ4fmjitY7K3iIn23Mnzy7OcEDhenfPIFVvhz4b1HTHuNUvlEC3cKLHCHDEjltzY4HXAH1rpvEjldDnjU4e42wr9XYL/AFrTjRYo1jXhVAA+grVVZKm6a2Zm6cXNT6odRRRWRoFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUVRvtYsdPISecGVvuwoNzt9FHNVPP1rUf8Aj3gXTYD/AMtJxvlI9kHA/E0AadzdW9nEZrmZIYx1Z2wKzf7Zur75dIsWlU/8vNxmOL6ju34CpbbQLOKUXFxvvbkf8trlt5H0HQfgK06AMcaE94Q+sXkl538lP3cI/wCAjr+JrUhhit4hFDGkaL0VFwBUlFABRRRQBkav++1LSrTs1wZm+iKT/Miteshf9I8WSN1W0tAv0Z2z/JRWvQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUVFcXVvaRGa5mSGMdWdgBQBLSEgAknAFY/8AbVzffLpFi8y/8/E+Y4vw7t+ApRoUl4Q+sXsl3/0wT93CPwHJ/E0APm8QWvmmCxjk1CccFLcZVf8AeboPzqP7HrGo83t2LGE/8sLQ5c/Vz/QVqwwQ20QigiSKNeiouAKkoAqWOl2OnA/ZbdUY/ec8s31Y8mrdFFABRRRQAUUUUAFFFR3Ey29vLM33Y0LH8BmgDM0T99c6nedfNuiin/ZQBf5g1r1meHYWh0G03j55E81vqx3H+dadABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUhOBk8AVlz+ILUSmCySTULgcFLcZC/7zdB+dAGrVK+1ex04hbicea33YkG52+ijmqf2TWNR5vLpbCE/8sbU5c/Vz/QVdsdKsdOB+y26ozfekPLt9WPJoApfaNa1Hi2t102A/8tbgb5SPZBwPxNS22gWcUouLkvfXI/5a3J3Y+g6D8BWpRQAUUUUAFFFFABRRRQAUUUUAFFFFABWV4kcroVxGpw8+2Ffq5C/1rVrJ1j99qWlWnZrgzN9EUn+ZFAGpGixRrGvCoAo+gp1FFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFUr7V7HTsLcTgSN92JRudvoo5qn9p1rUf+PW3XToD/AMtbkbpCPZBwPxNAGpcXMFpEZriZIYx1Z2AFZn9tXF98uj2Lzr/z8T5jiH0zy34CpLfQLSOUXF00l9cD/lrcndj6L0H4CtOgDIGhy3h3axevdD/nhH+7hH4DlvxNakEENtEIoIkijXoqKABUlFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVkL/pHixz1W0tAv0Z2z/JRWvWRon7661O86+bdGNT/soAv880Aa9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFULvUpLe9S0gs5LmRozIdjquACB/ER61bmmit4WmnlSKJBlndgqqPcmsm7mWSaPWbHUrAQCFojJId6HLA5BDD0q4WvqTK9tDQi1K0ksEvmlWGBh96UhdpzjBz0OeKcL+1a0ku0uI5IIwSzxsGHHXpWFb3Dz6amn6Tb3F0oyWvHcwIWJySCOTyTwKsyaHdSadJHLeyPL5TBIw7FNx7ncST/T0p2hf5/gK8rFhvEFku9V8ySSJN0scSFzH/snHG7PGKiEOpauqzPffZLOVQ6JbjEjKRkbmPQ/T86uxWcflySokkL3CfMjOSFJH93OAfpS2cNxapb2p2NDFbqhfGCWGB+WKTUbeY7u5Qshpmn3F3BZWUhmt4xJJIVJaTOeAzcseD7VrxSpNEksbBkdQykdwelVZoJIrue+jHmMbcRrEBySCx6++6pNPtjZ6dbWrNuaGJUJHcgUSUbaCV76lmiiioLCiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAjnlWC3kmb7saFj9AM1Q8OxNDoNpvHzyJ5rfVjuP86TxJIV0K4jU4efbCv1chf61pRxrFEka8KihR9BQA6iiigAooooAKKKKACiiigAooooAKKKKAOR+JOmXWo+Gg9th1tJhPLCTjzFAI4zwSCQQPb1xVH4f+EZrGK5vNXsURpnUwQyEMUwDliBkAnI9/lFdN4j/AHmnxWg63dxHD+BbJ/QGtatfay9n7Ppe5n7OPPz9ROlLRRWRoFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBk6x++1HSrTs1wZm+iKT/ADIrWrIX/SPFjnqtpaAfRnbP8lrXoAKKKKACiiigAooooAKKKKACiiigAooooAyL/wDf+ItMt+ohWS4YfQBR+rGtesiz/f8AibUJuot4o4F+pyx/mK16ACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKjnlWC3kmb7saFj9AM0AZmifvrrVLzr5t0Y1P+ygC/zzWvWZ4diaLQbTeMPInmt9WO7+tadABRRRQAUUUUAFFFFABRRRQAUUUUAFFFVdTufsel3VznHlRMw+uOKAKfh397a3N5/wA/V1JID/sg7R+i1rVS0e2+yaPaW56pCob645/WrtABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFZfiSRl0K4jQ4efbCv1chf61qVk6x++1HSrTs1wZm+iKT/ADIoA0441iiSNeFRQo+gp9FFABRRRQAUUUUAFFFFABRRRQAUUUUAFZPiP95p0doOt3cRw/gWyf0BrWrIv/3/AIi0y36iJZLhh9BtH6saANeiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACsO7uPL8SxSLDLcvHbtGkUW3IJKliSxA6bR+NbbMFUsxAAGST2rlIrJtdurjUIRZ3cK3UqKszEoQFRcgrnup/OrhyuXvEzvbQ6OHULaawjvfNEcEighpDtx7HNO+2WxtnuVnjeGMEs6MGAA69Kq22llNGi0+adiUGC8fGOcgDOeB0HsKS50s/2Jc2MEju0iNtMhGSew+lO0L79fwJvK23QdHrEc1+LNLeUybVZyWQbMjOCC2TxjOAetaFY8mnT3l5FclYI4jJHPkxYmUqB8pPf6+hIq8YJLyyjW5Z4JeGcQSFcH0yO1ElHSwRctbjP7VtzPdQoJJHtIw8m1frwPU8GrcciSxrJGwZHAZSO4PSqksMkN5cXyr5gNsqLGo+YlSx/XcKk022az022tnOWiiVGI9QKUlG10NN31LNFFFQWFFFFABRRRQAVk2n7/wATahN1FvFHAv1OWP8AMVrVk+Hf3trc3n/P1dSSA/7IO0fotAGtRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUVn3muWNnL5HmNPcHpBAu9z+A6fjQBoVXvNQtNPj8y7uI4V7bjyfoOprO/4nmpf3NKgP0kmP8A7Kv61Zs9DsbOXz/Lae4PWedt7n8T0/CgDm/F/wDa3ibw/La6Tp0/k71eQysIzOgOSqqeTng84Bxiqvw00LVtLlv7i8tpLO3mVFSCTALsM5faOnBA9/wFd9RWqqyVN0+jM3TTmp9UFFFFZGgUUUUAFFFFABRRRQAUUUUAFFFFAFXU7n7Hpd1c5x5UTMPqBxTNHtvsmj2dueqQqG+uOf1qt4k/eadHaDrd3EcP4Fsn9Aa1qACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKr3l/aWEXmXdxHCvbcev0Hes7+1NR1DjS7Axxn/l5vAUX6hPvH9KANd3WNC7sFUDJJOAKyn8QRzuYtKtpNQkBwWj+WJfq54/LNCaAk7CXVbmTUJAchH+WJfog4/PNaqIkaBI1CKowFUYAoAyP7M1LUOdTvzFGf+XazJUfQv1P4YrRs7C00+Py7S3jhXvtHJ+p71YooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigDIv8A9/4h0y36iISXDD6DaP1Y1r1k2n7/AMTX83UW8McC/U5Y/wAxWtQAUUUUAFFFFABRRRQAUUUUAFFFFABRRTXdY1LuwVQMkk4AoAdRWQ/iCKZzFpdvJqEgOC0fEa/Vzx+Wab/Zupahzqd95MR/5drMlR9C/U/higCzea3Y2cvkGQzXB6QQLvc/gOn41WzrmpdAmlQH1xJMR/6Cv61oWen2enx+XaW8cKnrtHJ+p6mrNAGdZ6HY2kvn7GuLnvPO29/zPT8K0aKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiqupXP2PTLq5zjyomYfUDigCn4e/e211ef8/V1I4P+yDtH6LWtVLRrb7Jo1nAeqQru+uOf1q7QAUUUUAFFFFABRRRQAUUVn3mt2NlJ5LSGW4PSCFd8h/AdPxoA0Kr3d9aWEXm3dxHCnYu2M/T1rO3a5qX3VTS4D3bEkxH0+6v61YtNDsbSXzyjXFz3nuG3v8AgT0/CgCv/amoahxpdgUjP/Lzd5RfqF+8f0py6Atwwk1a6k1BwchH+WJfog4/PNa9FADURIkCRoqKowFUYAp1FFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFZPiT95psdoOt3cRw/gWBP6A1rVkah+/8AEOmW/URCS4YfQbR+rGgDWpaKKACiiigAoqvd31rYRebd3EcKertjP09azv7Vv9Q40qwKxn/l5u8on1C/eP6UAa7MqKWdgqjkknAFZUniCKVzDpdvJqMoOCYuI1+rnj8s0i6Atywk1a6kv3HIjb5Yl+iDr+Oa1o40iQJGioi8BVGAKAMj+zdT1DnUr7yIj/y72ZK/m55P4YrQstOs9Pj8u0t0hB67RyfqepqzRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABWRafv/ABPfzdRbwxwL9Tlj/MVrdKyPD7K1pd3zEBbm6kkDE8bQdo/RaANiismTxBDLIYdMgk1GUHB8niNfq54/nTP7O1TUOdSvvs8R/wCXezJH5ueT+GKALV7rVjYyeS8pluD0ghXfIfwH9aq7tc1L7ippcB7viSYj6fdX9a0LLTrPToylpbpED1KjlvqepqzQBm2mhWNrL9oZWubnvPcNvf8ADPT8K0qKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigDkfiRZanfeH400+OWaNZw1zDCCWdMHHA5IDYJH+FZXw/8NXLWFw2sW1wlp5gNraTllUcfM3l9gTjr6E45r0OobqSSG1lkhiMsiISkY6sccCtVUfs/Z2W5m4Ln57hbNbFGjtTFsiYoVjxhCOowOh9qmrC0Wwv9LvBHcCKSKeEGR4QRiVTyzZPVt3/jtWrewvI9Ta4knDQlmITzZD16cFtv6UShFN2YKTa2NHehkMe4bwMlc8gev6Goory1mnkgiuInlj++iuCV+oqjALn7dqCwvEk/no2ZULAxbBjGCO4b9aqWsuy9lup9PuLaK1SQQRLEAMFgWYnPLMQCB/WhU0HOdBUX2mDeU85NyuEI3DhiMgfXHNR3a3xA+xyQJ8pyJUJycccg8fka59bK9s9O1KOaONZJZ4jC6OWMkuE+bkD+IZ/PsKIQUuoSm10OpooorI0CiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooATau7dgbsYzjmloooAKQqrYyAcHIyOlFFAC0UUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAf/Z)
Now,
Slope of line passing through (x1, y1) and (x2, y2) is
![](data:image/png;base64,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)
Slope of AB ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Slope of CD = ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
∴ Slope of AB = Slope of CD
Hence, AB is parallel to CD. … (1)
Now,
Slope of BC ![](data:image/png;base64,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)
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Slope of AD = ![](data:image/png;base64,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)
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![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACAAAAAXCAMAAABd273TAAAAAXNSR0IArs4c6QAAAEtQTFRFAAAAAAAAAAA6AABmADpmAGa2OgAAOjpmOmaQOpDbZgAAZjo6ZrbbZrb/kDoAkDo6kNv/tmYAttv/tv//25Bm/9uQ/9u2//+2///bgVTi6AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZklEQVQoU2NgGOxARACnC0XZGIGACZ8CXvzeE2WjogJBkGvAANlJomzsbIwsuN3IIMHPwyDGh8cXYC+IsnECSexWgBWIc7Hi9KsgSC/EBOxAEGi9GB+zME4FYtxAb3EIDfbEQqz7AI7/AxdxfCbLAAAAAElFTkSuQmCC)
∴ Slope of BC = Slope of AD
Hence, BC is parallel to AD. … (2)
From (1) and (2), we see that opposite sides of quadrilateral are parallel.
Find the angle of inclination of the straight line whose slope is
(i) 1 (ii) √3 (iii) 0
Answer:
If θ is the angle of inclination of the line, then the slope of the line is
m = tan θ where 0° ≤ θ ≤ 180°, θ ≠ 90°
i) tan θ = 1
⇒ θ = 45°
ii) tan θ = √3
⇒ θ = 60°
iii) tan θ = 0
⇒ θ = 0°
Question 2.
Find the slope of the straight line whose angle of inclination is
(i) 30° (ii) 60° (iii) 90°
Answer:
If θ is the angle of inclination of the line, then the slope of the line is m = tan θ
i) Given that m = tan 30°
⇒
ii) Given that m = tan 60°
⇒ m = √3
iii) Given that m = tan 30°
⇒
∴ The slope is undefined.
Question 3.
Find the slope of the straight line passing through the points
(i) (3, –2) and (7, 2) (ii) (2, –4) and origin
(iii) and
Answer:
Slope of straight line passing through the points (x1, y1) and (x2, y2) is given by
i) Slope of straight line passing through the points (3, –2) and (7, 2) is
ii) Slope of straight line passing through the points (2, –4) and (0, 0) is
iii) Slope of straight line passing through the points (1 + √3, 2) and (3 + √3, 4) is
Question 4.
Find the angle of inclination of the line passing through the points
(i) (1, 2) and (2, 3) (ii) and (0, 0)
(iii) (a, b) and (–a, –b)
Answer:
Slope of straight line passing through the points (x1, y1) and (x2, y2) is given by
i) Slope of straight line passing through the points (1, 2) and (2, 3) is
m = 1
If θ is the angle of inclination of the line, then the slope of the line is
m = tan θ where 0° ≤ θ ≤ 180°, θ ≠ 90°
∴ tan θ = 1 ⇒ θ = 45°
ii) Slope of straight line passing through the points (3, √3) and (0, 0) is
m = √3
If θ is the angle of inclination of the line, then the slope of the line is
m = tan θ where 0° ≤ θ ≤ 180°, θ ≠ 90°
∴ tan θ = √3 ⇒ θ = 30°
iii) Slope of straight line passing through the points (a, b) and (–a, –b) is
If θ is the angle of inclination of the line, then the slope of the line is
m = tan θ where 0° ≤ θ ≤ 180°, θ ≠ 90°
∴
Question 5.
Find the slope of the line which passes through the origin and the midpoint of the line segment joining the points (0, – 4) and (8, 0).
Answer:
We need to find slope of a line passing through origin (0, 0) & mid–point of P (0, –4) and Q (8, 0)
Let M mid–point of P (0, –4) and Q (8, 0)
Mid–point formula M (x, y)
Mid–point of PQ
We know that,
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of line between points O (0, 0) and M (4, –2)
Hence, slope of line is
Question 6.
The side AB of a square ABCD is parallel to x–axis. Find the
(i) slope of AB (ii) slope of BC (iii) slope of the diagonal AC
Answer:
i). Slope of AB
Since the side AB is parallel to x–axis,
Slope of side AB = 0
Therefore, the slope of AB is 0
ii). Slope of BC
The angle formed by the side BC is 90°
m = tan θ
θ = 90°
m = tan 90°
Therefore, the slope of BC is not defined.
iii). Slope of the diagonal AC
The diagonal AC is the angle bisector of the angle ∠BAD
∴ θ = 45°
m = tan θ
= tan 45°
= 1
Therefore, the slope of the diagonal AC is 1
Question 7.
The side BC of an equilateral 3ABC is parallel to x–axis. Find the slope of AB and the slope of BC.
Answer:
Slope of line BC:
The side BC is parallel to x–axis.
Slope of the side BC = 0.
Slope of line AB:
Since, this is an equilateral triangle each angle is 60°
∴ θ = 60°
m = tan θ
= tan 60°
= √3
Hence, the slope of AB is √3 and the slope of BC is 0
Question 8.
Using the concept of slope, show that each of the following set of points are collinear.
(2, 3), (3, –1) and (4, –5)
Answer:
(2, 3), (3, –1) and (4, –5)
Let A (2, 3), B (3, –1) and C (4, –5) be the given points
If the given point is collinear then,
Slope of AB = slope of BC
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of AB:
= –4
Slope of BC:
= –4
Slope of AB = Slope of BC = –4
Therefore, the given points are collinear.
Question 9.
Using the concept of slope, show that each of the following set of points are collinear.
(4, 1), (–2, –3) and (–5, –5)
Answer:
(4, 1), (–2, –3) and (–5, –5)
Let A (4, 1), B (–2, –3) and C (–5, –5) be the given points
If the given point is collinear then,
Slope of AB = slope of BC
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of AB:
Slope of BC:
Therefore, the given points are collinear.
Question 10.
Using the concept of slope, show that each of the following set of points are collinear.
(4, 4), (–2, 6) and (1, 5)
Answer:
(4, 4), (–2, 6) and (1, 5)
Let A (4, 4), B (–2, 6) and C (1, 5) be the given points
If the given point is collinear then,
Slope of AB = slope of BC
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of AB:
Slope of BC:
Therefore, the given points are collinear.
Question 11.
If the points (a, 1), (1, 2) and (0, b + 1) are collinear, then show that .
Answer:
Let A (a, 1), B (1, 2) and C (0, b + 1) be the given points.
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of AB
Slope of BC
If three points are collinear, then slope of AB is equal to slope of AC
∴ slope of AB = slope of BC
⇒ 1 = (–b + 1)(1 – a)
⇒ 1 = –b(1 – a) + 1(1 – a)
⇒ 1 = –b + ab + 1 – a
⇒ –b + ab – a = 0
⇒ ab – b = a
Dividing both sides by ab, we get
⇒
⇒
⇒
Hence proved.
Question 12.
The line joining the points A (–2, 3) and B (a, 5) is parallel to the line joining the points C (0, 5) and D (–2, 1). Find the value of a.
Answer:
Line joining points A (–2, 3) and B (a, 5) IA parallel to the line joining the points C (0, 5) and D (–2, 1)
The two lines are parallel only when their slopes are equal.
∴ slope of AB = slope of CD
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of AB is:
A (–2, 3) and B (a, 5)
Slope of CD is:
C (0, 5) and D (–2, 1)
As per the property, the two lines are parallel only when their slopes are equal.
i.e. Slope of AB = Slope of CD
⇒
⇒ 2 = 2(a + 2)
⇒ 2 = 2a + 4
⇒ 2a = 2 – 4
⇒ 2a = –2
⇒
⇒ a = –1
Question 13.
The line joining the points A (0, 5) and B (4, 2) is perpendicular to the line joining the points C (–1, –2) and D (5, b). Find the value of b.
Answer:
Line joining points A (0, 5) and B (4, 2) IA parallel to the line joining the points C (–1, –2) and D (5, b)
The two lines are perpendicular only if the multiplication of their slope is equal to 1.
∴ (Slope of AB) × (Slope of CD) = 1
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of AB is:
A (0, 5) and B (4, 2)
Slope of CD is:
C (–1, –2) and D (5, b)
The two lines are perpendicular only if the multiplication of their slope is equal to 1.
∴ (Slope of AB) × (Slope of CD) = –1
⇒
⇒
⇒
⇒ –b – 2 = –1 × 8
⇒ –b – 2 = –8
⇒ –b = –8 + 2
⇒ – b = – 6
⇒ b = 6
Question 14.
The vertices of ΔABC are A (1, 8), B (–2, 4), C (8, –5). If M and N are the midpoints of AB and AC respectively, find the slope of MN and hence verify that MN is parallel to BC.
Answer:
Given: vertices of triangle ABC i.e. A (1, 8), B (–2, 4), C (8, –5)
M and N are mid – points of AB and AC.
Finding co–ordinates of M and N:
We know that,
M is the mid–point of AB
x1 = 1, x2 = –2
y1 = 8, y2 = 4
Mid–point formula M (x, y)
Mid–point of AB
N is the mid–point of AC
x1 = 1, x2 = 8
y1 = 8, y2 = –5
Mid–point of AC
Slope of MN:
Slope of line passing through (x1, y1) and (x2, y2) is
Verification of MN and BC are parallel:
If MN and BC are parallel, then their slopes must be equal.
Slope of BC:
B (–2, 4) and C (8, –5)
Slope of BC
∴ Slope of MN = Slope of BC =
Hence, MN is parallel to BC.
Question 15.
A triangle has vertices at (6, 7), (2, –9) and (–4, 1). Find the slopes of its medians.
Answer:
Given: Vertices of triangle A (6, 7), B (2, –9) and C (–4, 1)
To find the slopes of medians. We need to know the mid–points of AB, BC and AC.
D, E and F are mid–points of AB, BC and AC respectively.
Mid – point formula =
Mid – point of AB =
Mid – point of BC =
Mid – point of AC =
F = (1, 4)
Slopes of median of triangles:
Slope of line passing through (x1, y1) and (x2, y2) is
A (6, 7) and E (–1, –4)
Slope of AE
B (2, –9) and E (1, 4)
Slope of BF
C (–4, 1) and D (4, –1)
Slope of CD
Question 16.
The vertices of a ΔABC are A (–5, 7), B (–4, –5) and C (4, 5). Find the slopes of the altitudes of the triangle.
Answer:
Let AD, BE and CF be the altitudes of a ΔABC.
Since, the altitude AD is perpendicular to BC,
Slope of BC
(slope of BC) × (Slope of AD) = –1 (∵ m1m2 = –1)
Let slope of AD be m1.
⇒
⇒
Since, the altitude BE is perpendicular to AC,
Slope of AC
(slope of AC) × (Slope of BE) = –1 (∵ m1m2 = –1)
Let slope of BE be m2.
⇒
⇒
Since, the altitude CF is perpendicular to AB,
Slope of AB
(slope of AB) × (Slope of CF) = –1 (∵ m1m2 = –1)
Let slope of CF be m3.
⇒
⇒
Question 17.
Using the concept of slope, show that the vertices (1, 2), (–2, 2), (–4, –3) and (–1, –3) taken in order form a parallelogram.
Answer:
Let A (–2, 2), B (1, 2), C (–1, –3) and D (–4, –3) be the given points taken in order.
Now,
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of AB
= 0
Slope of CD =
= 0
∴ Slope of AB = Slope of CD
Hence, AB is parallel to CD. … (1)
Now,
Slope of BC
Slope of AD =
∴ Slope of BC = Slope of AD
Hence, BC is parallel to AD. … (2)
From (1) and (2), we see that opposite sides of quadrilateral are parallel.
∴ ABCD is a parallelogram.
Question 18.
Show that the opposite sides of a quadrilateral with vertices A (–2, –4), B (5, –1), C (6, 4) and D (–1, 1) taken in order are parallel.
Answer:
Let A (–2, 2), B (1, 2), C (–1, –3) and D (–4, –3) be the given points taken in order.
Now,
Slope of line passing through (x1, y1) and (x2, y2) is
Slope of AB
Slope of CD =
∴ Slope of AB = Slope of CD
Hence, AB is parallel to CD. … (1)
Now,
Slope of BC
Slope of AD =
∴ Slope of BC = Slope of AD
Hence, BC is parallel to AD. … (2)
From (1) and (2), we see that opposite sides of quadrilateral are parallel.
Exercise 5.4
Question 1.Write the equations of the straight lines parallel to x- axis which are at a distance of 5 units from the x–axis.
Answer:Given
The line is ∥ to x axis that means it is a horizontal line at the fixed distance from x axis.
The equation of this line would be of the form y = k where k is some constant.
Here it is given the distance from x axis is 5 units which could be either in positive or negative direction, so the required equation would be
y = 5 or y = –5
Question 2.Find the equations of the straight lines parallel to the coordinate axes and passing through the point (–5,–2).
Answer:A line parallel to x axis is horizontal line and when this line passes through the given point (–5,–2) the equation of the line would be y = –2
When the line is parallel to the y axis, it is a vertical straight line passing through the point (–5, –2), the required equation of the line would be x = –5
Question 3.Find the equation of a straight line whose
(i) slope is –3 and y–intercept is 4. (ii) angle of inclination is 60° and y–intercept is 3.
Answer:The equation of the straight line with the given slope (m) and y –intercept ‘c’ is given the slope–intercept form
i.e. y = mx + c
(i) Here given slope = –3 (m) and y intercept = 4 (c)
So the required equation is
y = –3x + 4
⇒ 3x + y–4 = 0
(ii) Given angle of inclination = 60° and y intercept = 3 (c)
Here slope of the line (m) = tan θ = tan 60° = √3
The required equation is
y = mx + c
⇒ y = √3x + 3
⇒ √3x –y + 3 = 0
Question 4.Find the equation of the line intersecting the y- axis at a distance of 3 units above the origin and tan
, where θ is the angle of inclination.
Answer:Given
Angle of inclination = tan θ = 1/2 = slope of the line (m)
Also, the y intercept = 3 units (c) as the line is intersecting y axis at a distance of 3 units above origin
The equation of the straight line with the given slope (m) and y –intercept ‘c’ is given the slope–intercept form
i.e. y = mx + c
![](data:image/png;base64,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)
⇒ 2y = x + 6
⇒ x – 2y + 6 = 0
x – 2y + 6 = 0
Question 5.Find the slope and y–intercept of the line whose equation is
(i) y = x + 1 (ii) 5x = 3y (iii) 4x – 2y + 1 = 0 (iv) 10x + 15y + 6 = 0
Answer:The equation of the straight line with the slope (m) and y –intercept ‘c’ in the slope–intercept form is
y = mx + c
(i) here y = x + 1
⇒ slope of the line (m) = 1 and y–intercept ( c ) = 1
(ii) here 5x = 3y
3y = 5x
⇒ ![](data:image/png;base64,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)
Thus the slope of the line =
and y intercept ( c) = 0
(iii) 4x– 2y + 1 = 0
⇒ –2y = –4x –1
⇒ ![](data:image/png;base64,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)
Thus the slope of the line (m) = 2
and y –intercept (c) = 1/2
(iv) 10x + 15y + 6 = 0
⇒ 15y = = 10x – 6
⇒ ![](data:image/png;base64,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)
Thus the slope of the line (m) =
and y –intercept (c) = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAEhQTFRFAAAAAAAAAAA6OgAAOgA6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kLbbkNv/tmY625A627Zm29u2/7Zm/9uQ//+2///bhAIcygAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYUlEQVQYV42OyQ6AIAxEBxdcUBFk+f8/tbSGg8TEuby0M12At65FqRFI8wHfWXbjIHSa4ckulSAQgkGaFMk0674aJU36na9BR1PlL/fcqqS+fITYi5c3jbxbrvNJY2tz7gZDtgKX5jEBqQAAAABJRU5ErkJggg==)
Question 6.Find the equation of the straight line whose
(i) slope is –4 and passing through (1, 2) (ii) slope is
and passing through (5, –4)
Answer:The equation of the line with the given slope m and passing through the point (x1, y1) is given by the slope–point form
y– y1 = m (x – x1)
(i) Here given the slope (m) = –4 and the point = (1,2)
Thus the equation of the line is
y–2 = –4(x – 1)
⇒ y –2 = –4x + 4
⇒ y = –4x + 6
⇒ 4x + y – 6 = 0
(ii) Here given slope (m) = 2/3 and point = (5,–4)
Thus the required equation is
![](data:image/png;base64,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)
⇒ 3y + 12 = 2x –10
⇒ –2x + 3y + 22 = 0
⇒ 2x – 3y – 22 = 0
Question 7.Find the equation of the straight line which passes through the midpoint of the line segment joining (4, 2) and (3, 1) whose angle of inclination is 30°.
Answer:The equation of the line with the given slope m and passing through the point (x1, y1) is given by the slope–point form
y– y1 = m (x – x1)
here given angle of inclination = 30°
⇒ slope (m) = tan 30 ° = ![](data:image/png;base64,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)
Also the line passes through the midpoint of the line segment joining points (4, 2) and (3, 1)
⇒ the line passes through the point ( ![](data:image/png;base64,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)
The equation of the required line is
![](data:image/png;base64,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)
⇒![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK8AAAAZCAMAAABw344vAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjqQOmY6OmaQOma2OpCQOpDbZgAAZgBmZjoAZjpmZjqQZmYAZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A627Zm27aQ29u229v/2////7Zm/9uQ/9u2/9vb//+2///bU+VgFwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACMUlEQVRYR+1W21bCMBBMQKgoaBUvVdEoAaGtzf9/nrubJi1t0qtPHvLAIe1OZjK72ZSx8zg78M8ckLwY/bc2Dt2fj6mXjwEoAxmHHkKcPRyHwHLMOPQQ4ng+BGUw49DtzGp/z/lFuQLEM2Nqt6g8da9Uj+uDZnVyj+D8UMwYE/yJpeF0awPVK/wXk3eWRpPWQq7F9ULXyX3+ogssCeBXYPYlx7keyRWUb+2pZ6Fx6M40ml1aV+OSXnlrtInpVkWcz45JwJushji7nRN0ezViBJILpIGkF7bVsVlYVgbvfw4Y9JkXQbpZfkEhRzOwO26Qq+Pc6E6CabsCGFTUJJdJK4IKI91wLI3sjszKQs6X2NcoCdLbMkycB20Fg4H5qG6dyMmXIjeufarIiFARGv19oI0W/Wgf4ByXs3eAk5XiPOh2gzU58lyvbCG6Nmdr1gpPAoDKIicxhzm+jbEm/IPigNCBbtVbuFak2w0SRoSY52pUBAesdBmTAiwI6ib+oePc6EbL4KUlz9aPpWNbJ9MkqIcOFE3AcupmwI1znYEsvFkX5/90oXKcG91isCUHn7Kw6WI1TSi5hH6gBOmFot/lf+BUpNQa8EaxfaTKrvBKMXFudHNmDDkJ0HXlHraZ5UdIR0q+0KlPNwHnK10ncUOeynFudKNeQw5tfgL+cr9gd0fNyhezYer8BeNEtxTEyNfVRgtl1eOD1t+mR8rqDFfR6m3U52Vnpr8JhMKiW+48hjnwC7y/OI301h0tAAAAAElFTkSuQmCC)
⇒ 2x –2 √3 y + (3√3–7) = 0
Question 8.Find the equation of the straight line passing through the points
(i) (–2, 5) and (3, 6) (ii) (0, –6) and (–8, 2)
Answer:The equation of the line passing through the two given (x1, y1) and (x2,y2)points is given by form
![](data:image/png;base64,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)
(i) Here, (–2,5) and (3,6) are tow given points
The required equation is
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 5y –25 = x + 2
⇒ x–5y + 27 = 0
(ii) Here two points are (0, –6) and (–8, 2)
The equation is given by
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE0AAAAgCAMAAABgvQUGAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGaQAGa2OgAAOjoAOjo6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjpmZpDbZrbbZrb/kDoAkDpmkLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm29u229vb2////7Zm/7aQ/9uQ/9u2//+2///bpzzy/wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABKUlEQVRIS+VVDU+DMBR8RXHV1eHmR5WiHdoJ7f//gRZWaIS6PcoWY7yEhMC7e3dteQD8SZSbpZpn3PAnJ2CK2wgtQZJ8R4gT8WpygfVV0iSvaNr2rrMVQGGvFr2a4cvNDTLpLlWSuSAyVeb5rZHSGWnR+Kwo+3jnSH/y4qELou9y2Vnz3uxTANsIFXd77QvF1X1rbZCUKYP09vli+OWrU6hob817g3pNSLcYhw1u6cII0rnT3hoq16GiIu83dLYWgCDsBCq/KiH2R9EisUfnR/RVoZtjAYacY/XneY9Mep7m/0u1+dKRE3K4MGOq4av9KJ6OAFVnduoK5Lz93jFANZypulGcjhAVPyFdP/8nGVP1+tEO37ikY2pFo9VC1JJGn5AZ1Om7FMX4Agl5FfmxRXGVAAAAAElFTkSuQmCC)
⇒ –8y – 48 = 8x
⇒ x + y + 6 = 0 (dividing the whole equation by 8)
Question 9.Find the equation of the median from the vertex R in a ΔPQR with vertices at
P(1, –3), Q(–2, 5) and R(–3, 4).
Answer:The given points are
P (1,–3), Q(–2,5) and R (–3,4)
The median from R will pass through the midpoint of line PQ
(∵ Median passes through the mid–point of the side opposite to the vertex)
Let the mid–point of PQ be A = ![](data:image/png;base64,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)
The equation of the line passing through the two given (x1, y1) and (x2,y2)points is given by form
![](data:image/png;base64,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)
Thus here the required equation is
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 5y –20 = –6x –18
⇒ 6x + 5y –2 = 0
Question 10.By using the concept of the equation of the straight line, prove that the given three points are collinear.
(i) (4, 2), (7, 5) and (9, 7)
(ii) (1, 4), (3, –2) and (–3, 16)
Answer:The equation of the line passing through the two given (x1, y1) and (x2,y2)points is given by form
![](data:image/png;base64,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)
(i) The equation of the line passing through (4, 2), (7, 5)is
![](data:image/png;base64,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)
⇒ y–2 = x–4
⇒ x – y –2 = 0
Substituting the 3rd point in the equation of the line obtained
x–2 = y
⇒ 7 = 9–2
⇒ 7 = 7
Hence LHS = RHS
Since the third point satisfies the equation of the line obtained
⇒ All the three points lies in the same straight line or are collinear.
(ii) (1, 4), (3, –2) and (–3, 16)
The equation of the line passing through the points (1, 4), (3, –2) is
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ y–4 = –3x + 3
⇒ 3x + y –7 = 0
Substituting the 3rd point in the equation of line obtained
y = –3x + 7
⇒ 16 = –3 (–3) + 7
⇒ 16 = 16
LHS = RHS
Thus the three points are in the same straight line or are collinear.
Question 11.Find the equation of the straight line whose x and y–intercepts on the axes are given by
(i) 2 and 3 (ii) ![](data:image/png;base64,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)
(iii) ![](data:image/png;base64,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)
Answer:The equation of the straight line whose x and y intercepts are given as ‘a’ and ‘b’ respectively, is of the form
![](data:image/png;base64,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)
(i) Here given x intercept as 2 and y intercept as 3
The equation of the line
= ![](data:image/png;base64,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)
⇒ 3x + 2y = 6
⇒ 3x + 2y –6 = 0
(ii) Given x intercept as
and y intercept as ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYElEQVQYV42OORKAMAwDZQj3HUIgAf7/TYxTMECTbbawPBLwZckpaYFzmGATLVefifexuzVTKgZW1eOoNTwbi5K/WCgQG39yW0NUgHt/+2BKiVk+MybIsRzvrO463vnmAs3JA2tWswYWAAAAAElFTkSuQmCC)
The equation of the line
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ –9x + 2y = 3
⇒ –9x + 2y –3 = 0
(iii) Given x intercept as
and y intercept as ![](data:image/png;base64,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)
The equation of the line
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG4AAAAXCAMAAADdqIrAAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjpmOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLaQkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29v/2////7Zm/9uQ/9u2//+2///bHdXk+wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABj0lEQVRIS+1Ua1ODMBBMii1RQe1DK/UV21gkQP7/z/PuEhDaZkg76jiO94Fh4O52b/cSxv7j7yjw9vLds+RzgigFhxgNw23PIe3uSFbmVUwKqKlmPNpYuPugFmq0ZGzLb4KSmyQt0ncCSdbqKDiTxVgniWtoaAEUXezCSc4nheL88LANXGwyzAMLhvWvp0TRxt50EhqYzCet5igmpeCEehiN6S71Fu5C8DFuCvZRfm9y2CnCoELVIc5AGBc9EjJ6mnGeWvkdnFktWZVRWiku02b2vQ6lACYK+eJymdXwKoPs1wXLse4Tjt6pF3TzK2QlpCfaqANWxpbAdu3BWVPrxa3d1gPhbJckZLSRPYsPi+ngVNSFkzgXTQcC9XapB9pSBcL19Grho9UtIm7tdFY53Mcqgx8GgbX3HCs0wh1zGXbaSxFjCQhhHvHuOoM7qZrDS7IGR2Dv6in3dsJLbPxM7LVX874NJSymK/F4FPRZd09BUMXJSeBwyCk4uf9OocnSh58bDj1Ojrmmv2rMX9vnA09dIb3if1DqAAAAAElFTkSuQmCC)
⇒ 15x –8y –6 = 0
Question 12.Find the x and y intercepts of the straight line
(i) 5x + 3y – 15 = 0 (ii) 2x – y + 16 = 0
(iii) 3x + 10y + 4 = 0
Answer:The equation of the straight line whose x and y intercepts are given as ‘a’ and ‘b’ respectively, is of the form
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFkAAAAnCAMAAABT/RQsAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOmY6OmaQOma2OpCQOpC2OpDbZgAAZjoAZjo6ZjpmZmYAZmY6ZpDbZrb/kDoAkDo6kGYAkGY6kNv/tmYAtmY6tpA6ttv/tv/btv//25A627Zm27aQ2/+22////7Zm/9uQ/9u2//+2///b90xgHQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABe0lEQVRIS9VWYVPCMAxNhsJ0KBNUprOKVtxK+/9/n+nkptuNJsXjbuRrk9fk5TUNwBmaQpxWGnEdyt0V3sukmLxElKjI2xVBYCCHaQVQRwE3QXrBpaInHwB6xrl1z016PWcjTLoG9xTDhYfUgiJdMYPaMxJjdnXvS2WM6FBMM/oIVKPNeQJtfrPi7/8L7hR1r0a2haAEPp2sC9KozZEPqwWUcZQOn9c8Y0cAUy/iJSe6xxXzZ3nK22VA973nTnMjk2p5d4ehjrCD5GCpJts0k+CQHY/s3/R4kWmy7603d/Y5t8ctMUMRrRcKtHTGbAxXN1ZthH6Sfs46+K46U7lMqd0Xj4JO/7gE+y1GGXIcF7L7vEJk1wI9eS8xkbNHddN3Xzm/J4VNJ9kGdHhHG0LwewqHTHdL/vgejAC5mYwqapdx2yUpkM05HtnzDCfJudlcBcjezeb8wvPLs0kXsHvgO4gkoTdWQp3+vZKav3KOaH1bIl5y2vzXMx9B8Dfldh4jsrtZMQAAAABJRU5ErkJggg==)
(i) The given equation is
5x + 3y – 15 = 0
⇒ 5x + 3y = 15
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Hence the x intercept is 3 and y intercept is 5
(ii) The given equation is
2x – y + 16 = 0
⇒ 2x– y = –16
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Hence the x intercept is –8 and y intercept is 16.
(iii) Given equation is
3x + 10y + 4 = 0
⇒ 3x + 10y = –4
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Hence the x intercept is
and y intercept is ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAgCAMAAAA/gEgKAAAAAXNSR0IArs4c6QAAAEtQTFRFAAAAAAAAAAA6OgAAOgA6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kLbbkNv/tmYAtmY625A627Zm29u2/7Zm/9uQ//+2///b2hz91gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdElEQVQ4T8VSSRKAMAij7nuV2uX/LxU7elE41B7MqUMgZEIB8rH3StWMjO8WMIXmF7hKILDhBwy3glpRqFuq2/Gt5VtFYIj8IFIU1tNFREzoeqco/NqLt/GnC5SSlQmSEm4IrhTkwsR8kzBrfiJsFPPwJbADQh4C1RJQDjcAAAAASUVORK5CYII=)
Question 13.Find the equation of the straight line passing through the point (3, 4) and has intercepts which are in the ratio 3 : 2.
Answer:Given that the intercepts are in the ratio 3:2
Let the x–intercept be 3k and 2k, where k is some constant
Using the intercept –form, the equation of the line is
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
The point (3, 4) lies on this line, thus the point will satisfy the equation of the line
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 1 + 2 = k
⇒ K = 3
Thus the equation of the line is
⇒ ![](data:image/png;base64,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)
⇒ 2x + 3y = 18
⇒ 2x + 3y –18 = 0
Question 14.Find the equation of the straight lines passing through the point (2, 2) and the sum of the intercepts is 9.
Answer:Let the x intercept be ‘a’ and the y intercept be ‘b’ . it is given that
a + b = 9
The equation of the line using the intercept form is
![](data:image/png;base64,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)
Substituting b = 9– a in this equation
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ x (9–a) + ya = a(9 –a)
The point (2, 2) lies on this line and thus it satisfies the equation
x (9–a) + ya = a(9 –a)
⇒ 2(9 –a) + 2a = a ( 9–a)
⇒ 18 –2a + 2a = 9a –a2
⇒ a2 – 9a + 18 = 0
⇒ a2 – 3a –6a + 18 = 0
⇒ a ( a– 3) –6 (a –3) = 0
⇒ (a–6) (a–3) = 0
⇒ a = 6 or 3
The equation of the line is
x( 9–3) + y(3) = 3(9–3)
⇒ 6x + 3y = 18
⇒ 2x + y –6 = 0
Or
x(9–6) + y(6) = 6(9–6)
3x + 6y = 18
⇒ x + 2y –6 = 0
Question 15.Find the equation of the straight line passing through the point (5, –3) and whose intercepts on the axes are equal in magnitude but opposite in sign.
Answer:Let the x intercept be ‘a’. it is given the y intercept is equal in magnitude but opposite in sign
So y intercept = ‘–a’
The equation of the line using the intercept form is
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ x–y = a
The point (5, –3) lie son this line, thus it satisfies the given equation of the line
5 –(–3) = a
⇒ a = 8
The equation of the line is
x– y – 8 = 0
Question 16.Find the equation of the line passing through the point (9, –1) and having its x–intercept thrice as its y–intercept.
Answer:Let the x intercept be ‘a’. it is given that x intercept is thrice the y intercept
⇒ y intercept = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAeCAMAAADXVfSyAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OpDbZgAAZgA6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6tpBmtv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bDBke/AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZElEQVQYV42NRw6AMBADbXrvvRP+/0eSIHEAITGXkda2Fniyl2QIiChFay86XZWngJRejRqD1UPktDo6r/nXgRe/+3dx9GkkwFFUmORvxeZp72Wq1NDUBmY3g4hrbNIYXb17cAIGcQRNCycvzgAAAABJRU5ErkJggg==)
The equation of the line using the intercept form is
![](data:image/png;base64,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)
Substituting b = a/3 in the equation
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ x + 3y = a
the point (9,–1) lies on the equation of the line thus it must satisfies it
⇒ 9 + 3(–1) = a
⇒ a = 6
Hence the equation of the line is
x + 3y –6 = 0
Question 17.A straight line cuts the coordinate axes at A and B. If the midpoint of AB is (3, 2), then
find the equation of AB.
Answer:The mid–point of AB is in the 1st quadrant as both the intercepts are positive.
Thus A will lie on the y –axis and B will lie on the x axis
Let A be ( 0,a) and B be (b,0)
By the mid–point theorem, the coordinates of mid–point are
(
= (3,2)
⇒ ![](data:image/png;base64,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)
⇒
and![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADEAAAAeCAMAAACyuAbNAAAAAXNSR0IArs4c6QAAAGlQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6Ojo6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkGY6kLbbkNv/tmYAtmY6tpBmtv//25A627Zm27aQ29u22////7Zm/9uQ/9u2//+2///bgHGP9QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAv0lEQVQ4T72T3RaCIBCEd4vsjzJKKqxQeP+HjO3oqU55cLpob50PZoeR6B/T7pnXyEVBl3Sc3RAkaT1G1EtmiPCTipw6A66CYXXiIkvEy5ZZVVndU2C5pFZPATNWbDjeAZeI1MOE/erKcj8pxfdp5mKq+zzGXjRQm4iiGXiDQVe2ANvnpEoe8NUsUg7RAkRnFiBewrymhq3GhNtrwuZA9cdb5U54bAWNQ/epoTWk9CjgE+CRHyVoaS9CQBH9Jr4Dm1AHrSbMs44AAAAASUVORK5CYII=)
⇒ B = 6 and a = 4
Thus y intercept is 4 and x intercept is 6
The equation of the line using the intercept form is
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 4x + 6y = 24
⇒ 2x + 3y –12 = 0
Question 18.Find the equation of the line passing through (22, –6) and having intercept on x–axis exceeds the intercept on y–axis by 5.
Answer:Let the x intercept be ‘a’. It is given that x intercept exceeds the y intercept by 5
⇒ y intercept = a– 5
The equation of the line using the intercept form is
![](data:image/png;base64,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)
Substituting Value of b
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ x (a–5) + ya = a(a–5)
Given that the point (22, –6) lies on this equation of the line, hence it should satisfy it
22(a–5) + (–6) a = a(a–5)
⇒ 22a – 110 – 6a = a2 –5a
⇒ a2 – 21a + 110 = 0
⇒ a(a–10) – 11(a –10) = 0
⇒ (a–11) (a–10)
⇒ a = 11 or 10
The equation of the line is
x(11–5) + 11y = 11(11–5)
⇒ 6x + 11y – 66 = 0
Or
x(10–5) + 10y = 10(10–5)
⇒ 5x + 10y– 50 = 0
⇒ x + 2y –10 = 0
Question 19.If A(3, 6) and C(–1, 2) are two vertices of a rhombus ABCD, then find the equation of straight line that lies along the diagonal BD.
Answer:In a rhombus, the diagonals are perpendicular to each other and also bisect each other
Thus, the product of the slope of AC and BD will be –1 and BD will pass through the mid–point of AC
The points are given as A (3,6) and C ( –1, 2)
Slope of AC is
![](data:image/png;base64,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)
Thus, the slope of BD (m) = –1
The mid–point of AC is
![](data:image/png;base64,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)
Thus BD passes through the point (1,4)
Now using the slope point form, the equation of the line BD is:
(y –y1) = m (x – x1)
Here m = 1 and point is (1, 4)
⇒ y – 4 = (–1) (x –1)
⇒ y – 4 = –x + 1
⇒ x + y –5 = 0
Question 20.Find the equation of the line whose gradient is
and which passes through P, where P divides the line segment joining A(–2, 6) and B (3, –4) in the ratio 2 : 3 internally.
Answer:According to the section formula, the coordinates of the point P (x, y) with the given ratio m1 : m2 is
P = ![](data:image/png;base64,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)
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The gradient is the slope, thus the slope of the line is given as ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYElEQVQYV42OORKAMAwDZQj3HUIgAf7/TYxTMECTbbawPBLwZckpaYFzmGATLVefifexuzVTKgZW1eOoNTwbi5K/WCgQG39yW0NUgHt/+2BKiVk+MybIsRzvrO463vnmAs3JA2tWswYWAAAAAElFTkSuQmCC)
Using the slope –point form, the equation of the line is
y– y1 = m (x– x1)
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⇒ 2y – 4 = 3x
⇒ 3x – 2y + 4 = 0
Write the equations of the straight lines parallel to x- axis which are at a distance of 5 units from the x–axis.
Answer:
Given
The line is ∥ to x axis that means it is a horizontal line at the fixed distance from x axis.
The equation of this line would be of the form y = k where k is some constant.
Here it is given the distance from x axis is 5 units which could be either in positive or negative direction, so the required equation would be
y = 5 or y = –5
Question 2.
Find the equations of the straight lines parallel to the coordinate axes and passing through the point (–5,–2).
Answer:
A line parallel to x axis is horizontal line and when this line passes through the given point (–5,–2) the equation of the line would be y = –2
When the line is parallel to the y axis, it is a vertical straight line passing through the point (–5, –2), the required equation of the line would be x = –5
Question 3.
Find the equation of a straight line whose
(i) slope is –3 and y–intercept is 4. (ii) angle of inclination is 60° and y–intercept is 3.
Answer:
The equation of the straight line with the given slope (m) and y –intercept ‘c’ is given the slope–intercept form
i.e. y = mx + c
(i) Here given slope = –3 (m) and y intercept = 4 (c)
So the required equation is
y = –3x + 4
⇒ 3x + y–4 = 0
(ii) Given angle of inclination = 60° and y intercept = 3 (c)
Here slope of the line (m) = tan θ = tan 60° = √3
The required equation is
y = mx + c
⇒ y = √3x + 3
⇒ √3x –y + 3 = 0
Question 4.
Find the equation of the line intersecting the y- axis at a distance of 3 units above the origin and tan , where θ is the angle of inclination.
Answer:
Given
Angle of inclination = tan θ = 1/2 = slope of the line (m)
Also, the y intercept = 3 units (c) as the line is intersecting y axis at a distance of 3 units above origin
The equation of the straight line with the given slope (m) and y –intercept ‘c’ is given the slope–intercept form
i.e. y = mx + c
⇒ 2y = x + 6
⇒ x – 2y + 6 = 0
x – 2y + 6 = 0
Question 5.
Find the slope and y–intercept of the line whose equation is
(i) y = x + 1 (ii) 5x = 3y (iii) 4x – 2y + 1 = 0 (iv) 10x + 15y + 6 = 0
Answer:
The equation of the straight line with the slope (m) and y –intercept ‘c’ in the slope–intercept form is
y = mx + c
(i) here y = x + 1
⇒ slope of the line (m) = 1 and y–intercept ( c ) = 1
(ii) here 5x = 3y
3y = 5x
⇒
Thus the slope of the line = and y intercept ( c) = 0
(iii) 4x– 2y + 1 = 0
⇒ –2y = –4x –1
⇒
Thus the slope of the line (m) = 2
and y –intercept (c) = 1/2
(iv) 10x + 15y + 6 = 0
⇒ 15y = = 10x – 6
⇒
Thus the slope of the line (m) = and y –intercept (c) =
Question 6.
Find the equation of the straight line whose
(i) slope is –4 and passing through (1, 2) (ii) slope is and passing through (5, –4)
Answer:
The equation of the line with the given slope m and passing through the point (x1, y1) is given by the slope–point form
y– y1 = m (x – x1)
(i) Here given the slope (m) = –4 and the point = (1,2)
Thus the equation of the line is
y–2 = –4(x – 1)
⇒ y –2 = –4x + 4
⇒ y = –4x + 6
⇒ 4x + y – 6 = 0
(ii) Here given slope (m) = 2/3 and point = (5,–4)
Thus the required equation is
⇒ 3y + 12 = 2x –10
⇒ –2x + 3y + 22 = 0
⇒ 2x – 3y – 22 = 0
Question 7.
Find the equation of the straight line which passes through the midpoint of the line segment joining (4, 2) and (3, 1) whose angle of inclination is 30°.
Answer:
The equation of the line with the given slope m and passing through the point (x1, y1) is given by the slope–point form
y– y1 = m (x – x1)
here given angle of inclination = 30°
⇒ slope (m) = tan 30 ° =
Also the line passes through the midpoint of the line segment joining points (4, 2) and (3, 1)
⇒ the line passes through the point (
The equation of the required line is
⇒
⇒
⇒ 2x –2 √3 y + (3√3–7) = 0
Question 8.
Find the equation of the straight line passing through the points
(i) (–2, 5) and (3, 6) (ii) (0, –6) and (–8, 2)
Answer:
The equation of the line passing through the two given (x1, y1) and (x2,y2)points is given by form
(i) Here, (–2,5) and (3,6) are tow given points
The required equation is
⇒
⇒ 5y –25 = x + 2
⇒ x–5y + 27 = 0
(ii) Here two points are (0, –6) and (–8, 2)
The equation is given by
⇒
⇒ –8y – 48 = 8x
⇒ x + y + 6 = 0 (dividing the whole equation by 8)
Question 9.
Find the equation of the median from the vertex R in a ΔPQR with vertices at
P(1, –3), Q(–2, 5) and R(–3, 4).
Answer:
The given points are
P (1,–3), Q(–2,5) and R (–3,4)
The median from R will pass through the midpoint of line PQ
(∵ Median passes through the mid–point of the side opposite to the vertex)
Let the mid–point of PQ be A =
The equation of the line passing through the two given (x1, y1) and (x2,y2)points is given by form
Thus here the required equation is
⇒
⇒
⇒
⇒ 5y –20 = –6x –18
⇒ 6x + 5y –2 = 0
Question 10.
By using the concept of the equation of the straight line, prove that the given three points are collinear.
(i) (4, 2), (7, 5) and (9, 7)
(ii) (1, 4), (3, –2) and (–3, 16)
Answer:
The equation of the line passing through the two given (x1, y1) and (x2,y2)points is given by form
(i) The equation of the line passing through (4, 2), (7, 5)is
⇒ y–2 = x–4
⇒ x – y –2 = 0
Substituting the 3rd point in the equation of the line obtained
x–2 = y
⇒ 7 = 9–2
⇒ 7 = 7
Hence LHS = RHS
Since the third point satisfies the equation of the line obtained
⇒ All the three points lies in the same straight line or are collinear.
(ii) (1, 4), (3, –2) and (–3, 16)
The equation of the line passing through the points (1, 4), (3, –2) is
⇒
⇒ y–4 = –3x + 3
⇒ 3x + y –7 = 0
Substituting the 3rd point in the equation of line obtained
y = –3x + 7
⇒ 16 = –3 (–3) + 7
⇒ 16 = 16
LHS = RHS
Thus the three points are in the same straight line or are collinear.
Question 11.
Find the equation of the straight line whose x and y–intercepts on the axes are given by
(i) 2 and 3 (ii)
(iii)
Answer:
The equation of the straight line whose x and y intercepts are given as ‘a’ and ‘b’ respectively, is of the form
(i) Here given x intercept as 2 and y intercept as 3
The equation of the line
=
⇒ 3x + 2y = 6
⇒ 3x + 2y –6 = 0
(ii) Given x intercept as and y intercept as
The equation of the line
⇒
⇒
⇒ –9x + 2y = 3
⇒ –9x + 2y –3 = 0
(iii) Given x intercept as and y intercept as
The equation of the line
⇒
⇒
⇒
⇒ 15x –8y –6 = 0
Question 12.
Find the x and y intercepts of the straight line
(i) 5x + 3y – 15 = 0 (ii) 2x – y + 16 = 0
(iii) 3x + 10y + 4 = 0
Answer:
The equation of the straight line whose x and y intercepts are given as ‘a’ and ‘b’ respectively, is of the form
(i) The given equation is
5x + 3y – 15 = 0
⇒ 5x + 3y = 15
⇒
⇒
Hence the x intercept is 3 and y intercept is 5
(ii) The given equation is
2x – y + 16 = 0
⇒ 2x– y = –16
⇒
⇒
Hence the x intercept is –8 and y intercept is 16.
(iii) Given equation is
3x + 10y + 4 = 0
⇒ 3x + 10y = –4
⇒
⇒
⇒
Hence the x intercept is and y intercept is
Question 13.
Find the equation of the straight line passing through the point (3, 4) and has intercepts which are in the ratio 3 : 2.
Answer:
Given that the intercepts are in the ratio 3:2
Let the x–intercept be 3k and 2k, where k is some constant
Using the intercept –form, the equation of the line is
⇒
⇒
The point (3, 4) lies on this line, thus the point will satisfy the equation of the line
⇒
⇒
⇒ 1 + 2 = k
⇒ K = 3
Thus the equation of the line is
⇒
⇒ 2x + 3y = 18
⇒ 2x + 3y –18 = 0
Question 14.
Find the equation of the straight lines passing through the point (2, 2) and the sum of the intercepts is 9.
Answer:
Let the x intercept be ‘a’ and the y intercept be ‘b’ . it is given that
a + b = 9
The equation of the line using the intercept form is
Substituting b = 9– a in this equation
⇒
⇒ x (9–a) + ya = a(9 –a)
The point (2, 2) lies on this line and thus it satisfies the equation
x (9–a) + ya = a(9 –a)
⇒ 2(9 –a) + 2a = a ( 9–a)
⇒ 18 –2a + 2a = 9a –a2
⇒ a2 – 9a + 18 = 0
⇒ a2 – 3a –6a + 18 = 0
⇒ a ( a– 3) –6 (a –3) = 0
⇒ (a–6) (a–3) = 0
⇒ a = 6 or 3
The equation of the line is
x( 9–3) + y(3) = 3(9–3)
⇒ 6x + 3y = 18
⇒ 2x + y –6 = 0
Or
x(9–6) + y(6) = 6(9–6)
3x + 6y = 18
⇒ x + 2y –6 = 0
Question 15.
Find the equation of the straight line passing through the point (5, –3) and whose intercepts on the axes are equal in magnitude but opposite in sign.
Answer:
Let the x intercept be ‘a’. it is given the y intercept is equal in magnitude but opposite in sign
So y intercept = ‘–a’
The equation of the line using the intercept form is
⇒
⇒
⇒ x–y = a
The point (5, –3) lie son this line, thus it satisfies the given equation of the line
5 –(–3) = a
⇒ a = 8
The equation of the line is
x– y – 8 = 0
Question 16.
Find the equation of the line passing through the point (9, –1) and having its x–intercept thrice as its y–intercept.
Answer:
Let the x intercept be ‘a’. it is given that x intercept is thrice the y intercept
⇒ y intercept =
The equation of the line using the intercept form is
Substituting b = a/3 in the equation
⇒
⇒ x + 3y = a
the point (9,–1) lies on the equation of the line thus it must satisfies it
⇒ 9 + 3(–1) = a
⇒ a = 6
Hence the equation of the line is
x + 3y –6 = 0
Question 17.
A straight line cuts the coordinate axes at A and B. If the midpoint of AB is (3, 2), then
find the equation of AB.
Answer:
The mid–point of AB is in the 1st quadrant as both the intercepts are positive.
Thus A will lie on the y –axis and B will lie on the x axis
Let A be ( 0,a) and B be (b,0)
By the mid–point theorem, the coordinates of mid–point are
( = (3,2)
⇒
⇒ and
⇒ B = 6 and a = 4
Thus y intercept is 4 and x intercept is 6
The equation of the line using the intercept form is
⇒
⇒ 4x + 6y = 24
⇒ 2x + 3y –12 = 0
Question 18.
Find the equation of the line passing through (22, –6) and having intercept on x–axis exceeds the intercept on y–axis by 5.
Answer:
Let the x intercept be ‘a’. It is given that x intercept exceeds the y intercept by 5
⇒ y intercept = a– 5
The equation of the line using the intercept form is
Substituting Value of b
⇒ x (a–5) + ya = a(a–5)
Given that the point (22, –6) lies on this equation of the line, hence it should satisfy it
22(a–5) + (–6) a = a(a–5)
⇒ 22a – 110 – 6a = a2 –5a
⇒ a2 – 21a + 110 = 0
⇒ a(a–10) – 11(a –10) = 0
⇒ (a–11) (a–10)
⇒ a = 11 or 10
The equation of the line is
x(11–5) + 11y = 11(11–5)
⇒ 6x + 11y – 66 = 0
Or
x(10–5) + 10y = 10(10–5)
⇒ 5x + 10y– 50 = 0
⇒ x + 2y –10 = 0
Question 19.
If A(3, 6) and C(–1, 2) are two vertices of a rhombus ABCD, then find the equation of straight line that lies along the diagonal BD.
Answer:
In a rhombus, the diagonals are perpendicular to each other and also bisect each other
Thus, the product of the slope of AC and BD will be –1 and BD will pass through the mid–point of AC
The points are given as A (3,6) and C ( –1, 2)
Slope of AC is
Thus, the slope of BD (m) = –1
The mid–point of AC is
Thus BD passes through the point (1,4)
Now using the slope point form, the equation of the line BD is:
(y –y1) = m (x – x1)
Here m = 1 and point is (1, 4)
⇒ y – 4 = (–1) (x –1)
⇒ y – 4 = –x + 1
⇒ x + y –5 = 0
Question 20.
Find the equation of the line whose gradient is and which passes through P, where P divides the line segment joining A(–2, 6) and B (3, –4) in the ratio 2 : 3 internally.
Answer:
According to the section formula, the coordinates of the point P (x, y) with the given ratio m1 : m2 is
P =
The gradient is the slope, thus the slope of the line is given as
Using the slope –point form, the equation of the line is
y– y1 = m (x– x1)
⇒ 2y – 4 = 3x
⇒ 3x – 2y + 4 = 0
Exercise 5.5
Question 1.Find the slope of the straight line
3x + 4y – 6 = 0
Answer:Here we have the straight line : 3x + 4y –6 = 0
The slope intercept form is y = mx + b, where m is the slope and b is the y intercept.
Therefore, 3x + 4y –6 = 0
⇒ 4y + 3x = 6
⇒ 4y = 6–3x
⇒
(Divide both sides of the equation by 4)
⇒ ![](data:image/png;base64,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)
Now we will rewrite it in the slope intercept form
![](data:image/png;base64,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)
Hence according to the slope intercept form, y = mx + b
m i.e. slope is
.
Question 2.Find the slope of the straight line
y = 7x + 6
Answer:Here we have the straight line : y = 7x + 6
The slope intercept form is y = mx + b, where m is the slope and b is the y intercept.
Therefore when, y = 7x + 6
Hence according to the slope intercept form,
y = mx + b
m i.e. slope is 7.
Question 3.Find the slope of the straight line
4x = 5y + 3.
Answer:Here we have the straight line : 4x = 5y + 3
The slope intercept form is y = mx + b, where m is the slope and b is the y intercept.
Therefore, 4x = 5y + 3
⇒ –5y – 3 = –4x (Commutative law)
⇒ 5y + 3 = 4x (multiply by – on both sides of the equation)
⇒
(Divide both sides of the equation by 5)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Now we will rewrite it in the slope intercept form
![](data:image/png;base64,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)
Hence according to the slope intercept form, y = mx + b
m i.e. slope is
.
Question 4.Show that the straight lines x + 2y + 1 = 0 and 3x + 6y + 2 = 0 are parallel.
Answer:Given: Here the straight lines are x + 2y + 1 = 0 and
3x + 6y + 2 = 0.
To Prove: x + 2y + 1 = 0 and 3x + 6y + 2 = 0 are parallel.
Proof: If two lines are parallel then their slopes are equal.
Here slope of the first line x + 2y + 1 = 0 will be
![](data:image/png;base64,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)
(When the line is in the form ax + by + c = 0 then the slope of the line is
)
⇒ ![](data:image/png;base64,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)
Here slope of the second line 3x + 6y + 2 = 0 will be
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAK8AAAAuCAMAAABpoLk6AAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOjqQOmY6OmZYOmaQOma2OpCQOpC8OpDbZgAAZjoAZjpmZmYAZoFmZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tmZmttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bgsKVrAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACrklEQVRoQ+1ZaWPTMAy1A1sYR2FlHFsGSwZksITm//87nuRDTQ8Wz66hEH9wYleSn55lx1aVmsvRMjB81vqcq0e7EKcdNuxQnar2DVUv9ZWoduuNByyyicd7G4a3L4GSqxHEfXj7Z/WWfdbOVRjYdDa7YhvvdO3JTvVvtS6ISDyffyU1vBUfgbTUWr/nSnMIGMnON1gIrbtSL5S6haB+8o3HdbbYBKk2Wp/ct+Y9rvTlpfq5vMK841nRePT2w0cBM2Qjw0lyjxXqy7NatTwTnl+x5flt8ONQxcNVzck9O9ycEi8Ek3ponZhQELxOcixkXcGaErxiy+MdKvCbYOGtlsaIeVJt3gjcGK+TZKdEiPFy0+MVW2vR35cvEDTRZYyXaF0tKRApEn+L1wntxStTZEC2O5ZjOPx9/JKlB/nlpTWR39XFO7sYw0GuaTALKBRfZnDXs4nX94+FduAVWxIPw3W9Wpqh4kqrL9XwpYblc+wPBLqjnjveq9x64zeR5O51ob5E/FIFQzwxYstsCUNDAR7xXRcnsXs+vUHzO/bKBe8V6Ck+0HLDfslVj58wrpGUBglxi0IeeG55H6fibLE2z54uwG/MQSRuVmbtmYH/kwF/QDKnpL+/2APS2U2iL96hPZYD0vgwjyOnLUk+3MnckFvLlMvHphfeqUwv/iztTtXJeDiUoX38/hvxcCjWptu1FxZ7QJqu94ck+YAkVVIUc34nKZ2bxub8Tgp6c+R3cL9AfgerrIhGnCe/Y66guzJsoQ5kyu+0dJlv4y/IufI7nAS4rkPZ3JLPld+h7EVnU3UxoLPldxAQTYL0ZK78DtJyry9Mcjiu5MnvAGOTJLvjsjaHze8QpV2KdF/c3ARpd/GbWdB4McLYyVJsZjEQgnSHavHpiOil/OQr8zfJsZVf1ol/uVMNPH0AAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Now both the slopes are equal.
Hence both the lines are parallel.
Question 5.Show that the straight lines 3x – 5y + 7 = 0 and 15x + 9y + 4 = 0 are perpendicular.
Answer:Given: The straight lines are 3x – 5y + 7 = 0 and
15x + 9y + 4 = 0 .
To Prove: The straight lines are 3x – 5y + 7 = 0 and
15x + 9y + 4 = 0 are perpendicular.
Proof: If two lines are perpendicular, then the product of their slopes is equal to –1.
The slope of the first line,3x –5y + 7 = 0 is
(
)
The slope of the second line, 15x + 9y + 4 = 0
is
(
)
⇒ ![](data:image/png;base64,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)
Now the product of these slopes is m1×m2.
⇒ ![](data:image/png;base64,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)
As the product of the slopes is –1, the lines are perpendicular to
each other.
Question 6.If the straight lines
and ax + 5 = 3y are parallel, then find a.
Answer:Given: The straight lines are
and ax + 5 = 3y are parallel.
Since the lines are parallel their slopes should be equal.
Now,![](data:image/png;base64,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)
⇒ y = 2(x–p)
⇒ y = 2x –2p
⇒ m1
= 2
ax + 5 = 3y
⇒ 3y = ax + 5
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGEAAAAgCAMAAAA13OZYAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6ADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6Oma2OpDbZgAAZgA6ZjpmZpDbZrbbZrb/kDoAkDo6kDpmkGY6kLbbkLb/kNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bNxEumwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABX0lEQVRIS92VAVOCQBCFdxHyMrUyzOIoD4KA+/8/sD1AwMK7pWyccWecuXE+3ru3rC7ANVSyQO/xP4IoRPQiAL19gdQczl7qqZcsbiOQ5JcjDr4dtSx3iCtIhRcVIsistxo4lLsNQLleAcT0sVa13kBMynmQqaXdAEyXlrWaxBk5gAoy/bxn9IrUiZ5x3l0h2p58mEN1HylXBID0DtE4JAtHj+qr6pAUSRgaK3nz4IyQ00wofw+frzr0362B9fYgnIh2WgvhjlCF6L/hPBFzLessp0vHNKym/X1V7ghWxdAxh3HkHCT7FGiXg2wnizFM44jT4dfKhwenO0j6wTTF+0856dDpNAdHlGO6g8du06J/7k0vcMEunS3F9AxTra/BwZ550q7mwkNu0q7mwj84s6vZxYWHXL2rucWFh1y7q3kWXPgbV+9qbnHhjut3NcOCCx9z3a5mOAAX5nIczwszX4VvGgKKqiM9AAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Now as the lines are parallel ,m1 = m2
⇒ ![](data:image/png;base64,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)
⇒ 2×3 = a
⇒ a = 6
Question 7.Find the value of a if the straight lines 5x – 2y – 9 = 0 and ay + 2x – 11 = 0 are perpendicular to each other.
Answer:Given: The straight lines 5x – 2y – 9 = 0 and ay + 2x – 11 = 0 are perpendicular to each other.
As the lines are perpendicular to each other, the product of their slopes is equal to –1.
Slope of the first line 5x – 2y – 9 = 0 is m1.
(
)
Slope of the second line ay + 2x –11 = 0 is m2.
![](data:image/png;base64,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)
Therefore,![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
As the lines are perpendicular m1× m2 = –1
⇒ ![](data:image/png;base64,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)
⇒ a = 5
Question 8.Find the values of p for which the straight lines 8px + (2 – 3p)y + 1 = 0 and px + 8y – 7 = 0 are perpendicular to each other.
Answer:Given: The straight lines 8px + (2 – 3p)y + 1 = 0 and
px + 8y – 7 = 0 are perpendicular to each other.
Since the lines are perpendicular to each other, product of their slopes is equal to –1.
Slope of the first line 8px + (2 – 3p)y + 1 = 0 is m1.
i.e. ![](data:image/png;base64,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)
Slope of the second line px + 8y – 7 = 0 is m2.
i.e. ![](data:image/png;base64,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)
As the lines are perpendicular to each other m1× m2 = –1.
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAI0AAAAjCAMAAACelnkJAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmaQZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6tpA6ttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///b4ZH2AQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB6ElEQVRYR+2XbXuCIBSGoVrLrbL2Uq7VavSmwv//fQMMdYRyEPm08cUue87h5nDQR4T+h3sF8iXG06t7XJAIlsxRHs+D5HZPSuM1QrsJSvHjHg/egQnc1MCkXMaS2TUXRGS4RafBFhjppgYm5TLRNzPeN2R8RbJQLYPGWI41SA1nKJV0ueL1mcBoqvwQ9g40WaRohgdEHHbKRQ3nOkXFCSd8BwbtG1VL6qauAs8vsM4UtYcPN7XKy3t0+A2a5MtAQ1+L2Px2rRKZ1NZpsumBwGhSjHkra4MlDwKHxjPtD6PaSsMFQBpzKpbIc3/PCZnZpPGi4Yd/fFlOAB21K55E4ii09ami0eRlcMMPtTCW4Kqh7rTWEumQfrXhxanRWCe3Cvxo5E5Bzr7jTlmxTQLVxYDGgeX3qY2EESe8t0MFf/fcry5/K4qS3a6w9Teq2Ia/hfAIaqI8Z/MIvwhfU48/PgO9XwAnTRefv60e+9BuNK00lJPOnrTHprhhtbzKSXvsizGUaJ8K+WYlX2ztBrl00v3inETbVFYX7fBI0tgMsnLSvdIQ3SYgdI7Ex4OFpnTSfdKkHCatOU+62KJM0lgMcumke6Qpdqjug49RccKtllc56R5pmlN1s7yh0DpZ3lAwnS1vKKA/lvcHy48lx9HQlGwAAAAASUVORK5CYII=)
⇒ ![](data:image/png;base64,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)
⇒ p2 = –1×(2–3p)
⇒ p2 = 3p–2
⇒ p2–3p + 2 = 0
⇒ p.p–2p–p + 2 = 0
⇒ p(p–2)–1(p–2) = 0
⇒ (p–2)(p–1) = 0
⇒ p–2 = 0 ,p–1 = 0
⇒ p = 2,p = 1
Hence p = 1,2.
Question 9.If the straight line passing through the points (h, 3)and (4, 1) intersects the line 7x – 9y – 19 = 0 at right angle, then find the value of h.
Answer:Given: The straight line passing through the points (h, 3)
and (4, 1) intersects the line 7x – 9y – 19 = 0 at right angle.
As these lines intersect at right angle, they are perpendicular to each other.
When the lines are perpendicular to each other, the product of their slopes is equal to –1.
Slope of the first line passing through the points (h,3) and (4,1)
(here the two points are(x1,y1) and (x2,y2))
Now, ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Slope of the second line 7x – 9y –19 = 0 is
![](data:image/png;base64,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)
Therefore, product of the slopes is ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 14 = 36–9h
⇒ 9h = 36–14 = 22
⇒ ![](data:image/png;base64,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)
Question 10.Find the equation of the straight line parallel to the line 3x – y + 7 = 0 and passing through the point (1, –2).
Answer:Here it’s given that the straight line is parallel to the line
3x – y + 7 = 0 and passing through the point (1, –2).
As the lines are parallel to each other their slopes are equal.
Slope of the given line 3x – y + 7 = 0 is
![](data:image/png;base64,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)
Equation of the line passing through the point(1,–2) is
(y–y1) = m(x–x1),where (x1,y1) is (1,–2)
⇒ (y–(–2)) = 3(x–1)
⇒ y + 2 = 3x–3
⇒ y + 2–3x + 3 = 0
⇒ –3x + y + 5 = 0
⇒ 3x–y–5 = 0 (multiplied by –1 on both sides of the equation)
Hence the equation of the line is 3x–y–5 = 0 .
Question 11.Find the equation of the straight line perpendicular to the straight line x – 2y + 3 = 0 and passing through the point (1, –2).
Answer:Here it’s given that the straight line is perpendicular to the
straight line x – 2y + 3 = 0 and passing through the point (1, –2).
As the lines are perpendicular to each other , the product of their
slopes is equal to –1.
Slope of the given line x–2y + 3 = 0 is
![](data:image/png;base64,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)
Equation of the line passing through the point(1,–2) is
(y–y1) = m(x–x1),where (x1,y1) is (1,–2)
)
⇒ 2(y + 2) = (x–1)
⇒ 2y + 4 = x–1
Question 12.Find the equation of the perpendicular bisector of the straight line segment joining the points (3, 4) and (–1, 2).
Answer:Given: There is a perpendicular bisector of the straight line segment joining the points (3, 4) and (–1, 2).
We have to find the equation of the perpendicular bisector.
As it is perpendicular to the given line segment,the product of their slopes is equal to –1 and as it bisects the line segment,it implies it divides the line segment into 2 equal parts.
Thus,mid–point of the line segment joining the points (3, 4) and (–1, 2) is:
; where (x1,y1) and (x2,y2) are the end points of the line segment
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= 1,3
Therefore the mid–point is (1,3).
The slope of the line segment joining the points (3, 4) and
(–1, 2) is:
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Therefore the equation of the perpendicular bisector is
(y–y1) = m(x–x1)
Now substitute the value of the mid–point(1,3) and slope
in the above equation.
![](data:image/png;base64,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)
⇒ 2(y–3) = (x–1)
⇒ 2y –6 = x–1
⇒ 2y–6–x + 1 = 0
⇒ –x + 2y–5 = 0
⇒ x–2y + 5 = 0 (multiply by –1 on both the sides of the equation)
Question 13.Find the equation of the straight line passing through the point of intersection of the lines 2x + y – 3 = 0 and 5x + y – 6 = 0 and parallel to the line joining the points (1, 2) and (2, 1).
Answer:Here it is given that the straight line passing through the point of intersection of the lines 2x + y – 3 = 0 and 5x + y – 6 = 0 and parallel to the line joining the points (1, 2) and (2, 1).
As the straight line passes through the point of intersection of the lines 2x + y – 3 = 0 and 5x + y – 6 = 0 , we should find the intersection point by solving these equations:
![](data:image/png;base64,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)
i.e –3x = –3
⇒ x = 1
Substitute x = 1 in 2x + y–3 = 0
We get,2(1) + y–3 = 0
⇒ y–1 = 0
⇒ y = 1
Therefore , the intersection point is (1,1).
As the line passing through(1,1) is parallel to the line segments
joining the points (1, 2) and(2, 1),their slopes are equal.
Slope of the line joining the points (1, 2) and(2, 1) is
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Hence the equation of the line passing through the point (1,1) with slope m equal to –1 is
(y–y1) = m(x–x1)
⇒ (y–1) = –1(x–1)
⇒ y–1 = –x + 1
⇒ y–1 + x–1 = 0
⇒ x + y–2 = 0
Question 14.Find the equation of the straight line which passes through the point of intersection of the straight lines 5x – 6y = 1 and 3x + 2y + 5 = 0 and is perpendicular to the straight line 3x – 5y + 11 = 0.
Answer:Here we have the straight line which passes through the point of intersection of the straight lines 5x – 6y = 1 and 3x + 2y + 5 = 0 and is perpendicular to the straight line 3x – 5y + 11 = 0.
As the straight line passes through the point of intersection of the lines 5x – 6y = 1 and 3x + 2y + 5 = 0, we should find the intersection point by solving these equations:
5x –6y = 1
3x + 2y + 5 = 0
⇒ 5x – 6y– 1 = 0 –––(1)
and 3x + 2y + 5 = 0–––(2)
Now multiply equation (2) by 3 on both the sides.
Thus we have 9x + 6y + 15 = 0–––(3)
Now, we have
![](data:image/png;base64,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)
⇒ 14x = –14
⇒ x = –1
Now, substituting x = –1 in the equation 5x–6y = 1, we have,
5(–1)–6y = 1
⇒ –5–6y = 1
⇒ –6y = 1 + 5 = 6
⇒ ![](data:image/png;base64,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)
⇒ y = –1
Therefore the point of intersection is (–1,–1).
Slope of the line 3x –5y + 11 = 0 is :
![](data:image/png;base64,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)
As the line which passes through (–1,–1) is perpendicular to 3x –5y + 11 = 0 ,the product of their slopes will be –1.
Therefore,![](data:image/png;base64,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)
⇒
(Here we have multiplied
on both the sides)
⇒ ![](data:image/png;base64,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)
Hence the equation of the line passing through (–1,–1) and slope as
is:
(y–y1) = m(x–x1)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 3y + 3 = –5x–5
⇒ 3y + 3 + 5x + 5 = 0
⇒ 5x + 3y + 8 = 0
Question 15.Find the equation of the straight line joining the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4 and the point of intersection of the lines 2x + y – 4 = 0 and x – 2y + 3 = 0.
Answer:Here we have the straight line which joins the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4 and the point of intersection of the lines 2x + y – 4 = 0 and x – 2y + 3 = 0.
As the straight line which joins the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4,let us solve these 2 equations:
3x –y + 9 = 0–––(1)
X + 2y–4 = 0–––(2)
⇒ 3x –y + 9 = 0–––(1)
3x + 6y–12 = 0–––(2)(multiply (2) equation by 3 on both the sides)
Now,
(Subtract equation (2) from (1))
![](data:image/png;base64,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)
⇒ –7y = –21
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGAAAAAgCAMAAADaHo1mAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOmZmOmaQOma2OpCQOpDbZgAAZgA6ZjoAZjpmZrb/kDoAkDo6kDqQkGYAkGY6kLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bHQFk9wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABNElEQVRIS+VVTVfCMBDcBZGooHwFBaKGQtM2//8HupP28R4HJWD34HMu7aHZ2dmZbYj+D44L5inkxvexhupmvqVisCPaPy1UCNB09SAERF6NwL+k2agRFMkCPQLf1ddSEKR+WCFF7r5UiFEzY4EQeDxbM35CLB6Z71IodOAGW6otcq0Ehyx7kWyZR2VlWIXLDT8p2pFYFjTq15vJB2QIy3nyHGxMyKH97mukYoK4VWZFcX3RjBNp+5Ll3MGg+WjHFDAmBYSUZ5mRwwKd0M+IUK4yIGhmz0tR0i+iRemAzSRyGYt5LX3EjtUpocIDK/pGvTFyA7bWBrULBNXXu4yMdvKirD3zle3Y6Vv2ifha5qzM+bRtu27ZwLWgifx53tiFbiDkv3L5p3Vj592xPy/gd/J1Tn8B7VoSdzL/x+AAAAAASUVORK5CYII=)
Substitute y = 3 in the first equation
3x–y + 9 = 0
⇒ 3x–3 + 9 = 0
⇒ 3x + 6 = 0
⇒ 3x = –6
⇒ ![](data:image/png;base64,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)
Thus, the point of intersection is(–2,3).
Now,
2x + y – 4 = 0–––(3)
x – 2y + 3 = 0–––(4)
(multiply (4) equation by 2 on both the sides) we get
2x – 4y + 6 = 0 --------(5)
Now,
![](data:image/png;base64,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)
⇒ 5y = 10
⇒ ![](data:image/png;base64,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)
Substitute y = 2 in the (3) equation:
2x + y–4 = 0
⇒ 2x + 2–4 = 0
⇒ 2x–2 = 0
⇒ 2x = 2
⇒ ![](data:image/png;base64,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)
The point of intersection is (1,2).
Hence equation of the line is
![](data:image/png;base64,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)
Here we have the points (–2,3) and (1,2)
Therefore,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ 3(y–3) = –1(x + 2)
⇒ 3y–9 = –x–2
⇒ 3y–9 + x + 2 = 0
⇒ 3y + x–7 = 0
⇒ x + 3y–7 = 0
Question 16.If the vertices of a Δ ABC are A(2, –4), B(3, 3) and C(–1, 5). Find the equation of the straight line along the altitude from the vertex B.
Answer:![](data:image/jpeg;base64,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)
Given: A ∆ABC has vertices A(2,–4),B(3,3) and C(–1,5).
The straight line BD drawn from vertex B is perpendicular to the line AC. So the product of the slopes of BD and AC is equal to –1.
Slope of AC is m1.
![](data:image/png;base64,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)
Where (x1,y1) and (x2,y2) are (2,–4) and (–1,5)
Therefore,![](data:image/png;base64,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)
Slope of the BD is ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAgCAMAAADDlWPAAAAAAXNSR0IArs4c6QAAAE5QTFRFAAAAAAAAAAA6ADqQAGaQAGa2OgAAOmZmOpDbZgAAZjoAZrb/kLbbkLb/kNv/tmYAtmY6tpA6tpBm25A625Bm2////7Zm/9uQ//+2///bCAd+fQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAbElEQVQoU2NgIA1ICrCiahBh50ITYWAQxiEiwckIBtxAI3CpQTJdkp9FFMUyYZBmDtIcTJpqiPuQAGna6aJajJFZiI2JR5CRiRdmnzATHwM/CMMDXphZlAGMiRARgAWzOBsjqxgwhEGYSK8AAEazA9GN3Y3BAAAAAElFTkSuQmCC)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bjLFCZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAW0lEQVQYV42MSQ6AIBAEG1TcV1BUlP8/U2YOHiQa61JJp2aACD9ntNmyYQPLt71Jd6pEoIrfvSxUB373d2gLIVvADxNWqXl2OfsYO5IRCRvYVI+z1nDBsIrvHlxvGQNJwgEIiAAAAABJRU5ErkJggg==)
Hence the equation of the line BD drawn from the vertex B is
(y–y1) = m(x–x1),here B is (3,3) and m = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6ADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ2////9uQ/9u2//+2///bjLFCZQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAW0lEQVQYV42MSQ6AIBAEG1TcV1BUlP8/U2YOHiQa61JJp2aACD9ntNmyYQPLt71Jd6pEoIrfvSxUB373d2gLIVvADxNWqXl2OfsYO5IRCRvYVI+z1nDBsIrvHlxvGQNJwgEIiAAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ 3(y–3) = (x–3)
⇒ 3y –9 = x–3
⇒ 3y–9–x + 3 = 0
⇒ –x + 3y–6 = 0
⇒ x–3y + 6 = 0
Question 17.If the vertices of a Δ ABC are A(–4,4 ), B(8 ,4) and C(8,10). Find the equation of the straight line along the median from the vertex A.
Answer:![](data:image/jpeg;base64,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)
Given: The ∆ABC with vertices A(–4,4),B(8,4) and C(8,10) and
AD is the median,i.e. it passes through the mid point of BC.
Therefore D is midpoint of BC where B(8,4) and C(8,10).
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
Equation of AD is
,where (x1,y1) and (x2,y2) are (–4,4) and (8,7)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAG0AAAAkCAMAAAC0cUTAAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgBmOjoAOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kDpmkDqQkGaQkLb/kNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/7aQ/9uQ/9u2//+2///bGv2u4wAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACAklEQVRYR+2W61LCMBCFE1QkogVBhXprkYK2lub9385N0iab2JJQZ2Ac3R+gbfac7ObCR8gfi/e7t7rizX3ep3YjYLK7pLKJGFOypfi87mGnBEzsk1IG/HEs3IibGVCqO0MkldJBUlAqlVUo/WwRy2fl1dpjsGWDpGRD0wN3gkaK7KYgvkKlc+lSROobakx85RTDPIuMWZ2os5AUFDHM+dOaVFMqYykHV/M1jxcvIOHmtlln5w+QYgs0/1lSMOo2sUqX8pl0Ft0JcduMUR/dDCwFU00v53hpGvnwTn6+8viiOTNt89NScqvbG7au9IOdiUn4d8mGjXgq+2BtM9RyLQXPYI2sxbD1+5yAjm0MUqsEb0h1BCLj3u90IwFbKqXfXx315vKdryO9T9WRhBh4j3z7lLRA5x++UvwKeoRP6oTvf97JE07+3zqgAy1gYriEx2KTjuzrUnJJSOxmlN5Y+OJcrg6X8Ofc/ZFuYMJvx+OJIgAdzuXahjiFfV82MOF3q6bQhBQ3xikNcUmj5vCHhgm/G4+jfCcctZQimiYsLqkfFmJymi0Ml/jdiFg3oKAOMMGI01GagQm/WzVbAIugTtoo43CJ1JOlWRGCPzKhZHvdYATmEpHRQo01TPhrI1vmnACXLjCXBOgdOKSDnHogTpAxJhud0Atxguz23lxBCr9u0BdNcje+ktxbigAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 12(y–4) = 3(x + 4)
⇒ 12y–48 = 3x + 12
⇒ 12y–48–3x–12 = 0
⇒ –3x + 12y–60 = 0
⇒ 3x–12y + 60 = 0
⇒ x–4y + 15 = 0(Divide both sides of the equation by 3)
Hence Equation of AD is x–4y + 15 = 0.
Question 18.Find the coordinates of the foot of the perpendicular from the origin on the straight line 3x + 2y = 13.
Answer:Here we have a perpendicular from the origin i.e.(0,0) to the
straight line 3x + 2y = 13.
We have to find the foot of the perpendicular i.e the intersection point at the line 3x + 2y = 13.
As these lines are perpendicular the product of their slopes is equal to –1.
Slope of the 3x + 2y–13 = 0 is m.
![](data:image/png;base64,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)
Therefore the slope of perpendicular is ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAgCAMAAAA2a+hwAAAAAXNSR0IArs4c6QAAAEVQTFRFAAAAAAAAAAA6ADqQAGaQAGa2OgAAOmZmOpDbZjoAkLbbkLb/kNv/tmYAtmY6tpA6tpBm25A625Bm2////7Zm/9uQ///bKPmyRAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAa0lEQVQ4T2NgoCoQ42PFaZ4QOyduSQYGQfpLivGyCONyriAjEHBQNXCoZhgvyG1gwMQDNBTOAzKoZsdQM0iEkVmAjYmLn5GJG4vTBYEBxQvC2BKZILMwAxiTLcmHJR2JsjGyigCTEAiTH5gAb0UDaybakt8AAAAASUVORK5CYII=)
i.e.![](data:image/png;base64,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)
Hence the equation of the perpendicular from (0,0) and slope as ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAgCAMAAADpJZJvAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQYV42OyQ6AIAxEp4j7vqCC+P+/aYHEA8TEuby0M20GiHW2RCVgmxm7WLxr8sCt8tjZdlOAZugBtibWkLz7Wrg063f+DaqCRAfcY9LvmnqXWinzBA7JvZoFhgkl/V2kB6PpA2t1TzLqAAAAAElFTkSuQmCC)
is (y–y1) = m(x–x1)
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 3y = 2x
⇒ 3y–2x = 0
⇒ 2x–3y = 0
Now solve the two equations 3x + 2y–13 = 0 and 2x–3y = 0.
3x + 2y–13 = 0–––(1)
2x–3y = 0–––––(2)
Multiply (1) by 3 and (2) by 2 and add
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Substitute x = 3 in the equation 2x–3y = 0.
2(3)–3y = 0
⇒ –3y = –6
⇒ y = 2(Divide both the sides of the equation by –3)
Hence the coordinates of the foot of the perpendicular is(3,2)
Question 19.If x + 2y = 7 and 2x + y = 8 are the equations of the lines of two diameters of a circle, find the radius of the circle if the point (0, –2) lies on the circle.
Answer:Given: The equations of the lines of two diameters of a circle
are x + 2y = 7 and 2x + y = 8 and F(0,–2)
![](data:image/jpeg;base64,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)
Now to find the center of the circle,we have to find the
intersection of the lines x + 2y = 7 and 2x + y = 8
x + 2y = 7–––(1)
2x + y = 8–––(2)
Multiplying (1) by 2,we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Substitute y = 2 in x + 2y = 7 we get,
x + 2(2) = 7
⇒ x = 7 – 4 = 3
The point of intersection is (3,2) i.e.the center.
Therefore distance between the points (3,2) and (0,–2) is
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= 5 units
Hence the radius of the circle is 5 units.
Question 20.Find the equation of the straight line segment whose end points are the point of intersection of the straight lines
2x – 3y + 4 = 0, x – 2y + 3 = 0 and the midpoint of the line joining the points (3, –2) and (–5, 8).
Answer:
![](data:image/jpeg;base64,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Here its given that the straight line segment has end points as the point of intersection of the straight lines
2x – 3y + 4 = 0, x – 2y + 3 = 0 and the midpoint of the line joining the points (3, –2) and (–5, 8).
As one end point is point of intersection of the straight lines
2x – 3y + 4 = 0, x – 2y + 3 = 0,we will solve these equations.
2x–3y + 4 = 0––––(1)
x–2y + 3 = 0–––––(2)
Multiply (2) by 2,then we have
![](data:image/png;base64,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)
i.e. y = 2
Substitute y = 2 in 2x–3y + 4 = 0
2x–3(2) + 4 = 0
⇒ 2x–6 + 4 = 0
⇒ 2x–2 = 0
⇒ 2x = 2
⇒ x = 1
Therefore the lines intersect at (1,2).
Now the line has one end point as (1,2) and other end point as mid– point of the line joining (3, –2) and (–5, 8).
The mid– point of (3, –2) and (–5, 8) is
![](data:image/png;base64,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)
= ![](data:image/png;base64,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)
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACIAAAAgCAMAAABAUVr7AAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6ADqQAGa2OgAAOgA6Ojo6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZrbbZrb/kDoAkLbbkNv/tmY625A627Zm29u229vb2////7Zm/9uQ/9u2//+2///b7LW9vQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAwklEQVQ4T62SWxeCIBCEsauUZlGCqcH//5dyq7MDPuQ57OMwzC7Lx1jhGm5VxTFTdfVIFN0+mdq9iGL6CzWEk/lELfKwMqe80hBRd2fXSDeVr4c9VDDKfOTvQWCSxGF1a7vKPR1nso7JhcUygo8GUkJHYmEft4X8TYV3XTwuLI7Ur0PUinfcFohrX737h0U39KtiCnLb370MYs6tdeQichvTE5gpt99XAMzIbbSAmHAbLCCm3HoHiDm37kUZzNv+CN0Li1YJTsmcSBcAAAAASUVORK5CYII=)
= (–1,3)
Hence to find the equation of the line with end points as (x1,y1) and (x2,y2) ,we use:
,where (x1,y1) and (x2,y2) are (1,2) and (–1,3)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ –2(y–2) = (x–1)
⇒ –2y + 4 = x–1
⇒ –2y + 4–x + 1 = 0
⇒ –2y–x + 5 = 0
⇒ x + 2y–5 = 0(multiply by –1 on both the sides of the equation)
Question 21.In an isosceles Δ PQR, PQ = PR. The base QR lies on the x–axis, P lies on the y– axis and 2x – 3y + 9 = 0 is the equation of PQ. Find the equation of the straight line along PR.
Answer:Here it’s given that in an isosceles
PQR, PQ = PR. The base QR lies on the x–axis, P lies on the y– axis and 2x – 3y + 9 = 0 is the equation of PQ.
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCAEPAQ0DASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aiiigAooooAKKKKACiiigAooooAKKKKACiiigAoorD8WymHSIm8yVFN1CrmIkMVLDIGOenpVQjzSUSZS5Ytm5RXM6cD/AG3bf2WNQFpsf7V9q8zZ0+XG/ndn07V01OcOVihLmQUUUVBYUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAEc08NtE008qRRr953YKB9SapTx2WvWkRgvElijnSUPA4YblOQMiuA+LT3H9p6ZFO2LBonKKT8jTAjr6kL0z71lfDeWWLxvFDZuFjkgkN2in5SoHykj13dO+M11xoP2PtlLY5pVl7X2TR7LRRRXIdIUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABVe9vYNPtJLq4cJHGMk+vsPepndY0Z3YKqjJJPAFc3Aj+KdRF3KpGlWzfuYyP8AXuP4j7UAZ9/btfaRe65rECP5kRS0tpV3LGp4Bwe/Oah8L6PZ6h4fMFoiWd5ZykxTRKFIzyM46jqK6rXdLTVdNaB5XjVDv+THJAOBWT4I01INNW/WaRnuVw6HG0YY4IoA0NH1mS4lbTtRQQahCPmXtIP7y1sVm6xo0eqxKyuYLqE7oZ16of8ACoNH1mSaZtN1JBDqEQ5HaUf3loA2aKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiisHW9QnuLldF0xv8ASpR++lHSFO5+tAEF9NJ4j1BtLtXK2EB/0uZf4z/cBrooYY7eFIYkCRoNqqOgFQadp8Gl2UdrbrhEHJPVj3J96tUARzjNvIP9g/yrH8HH/imrf2Z//QjWzLzE4/2TWL4N/wCRci9nf/0I0AbtZusaPFqsKkMYbmI7oZ1+8h/wrSooAxdH1mWSdtM1NRDqEQ/4DMP7y1tVnaxo8OrQLljFcRHdDOv3kP8AhVXSNYma4Ol6ooiv4xwf4Zh/eWgDbooooAKKKKACiiigAooooAKKKKACiiigAooooAKKa4YxsEYKxB2sRnB+lcXb2k1h4k06zsdXvtQ1FW36s8kxaERlT95M7UJbG0Lg4z2oA7aiuZ03xPfX+uHRGsEjvLMsdQbcdkaf8syh/iL8EegBzW1e6tpumvGl9f21q0v3BNKqFvpk89aALlFczN4qnhhuoDaRnUYtRSyih3Ha+8go+fTYSx/3TUN541+y+JodNGmai0JilLlbJyzMrKAV9V5PP0oA6yivOtfv3hv9bmvtQ1O1v7dPM0mG33iNkEYIIAG18tuDbugHatW48X3sKWxEmhRmW2ilZbzUvJkBZQT8u04HPHNAHYUVxxkm8Ta7bWNzeNDbJpoumGnXbBJJGcqCJFwWAC8e5qzoviBrbwkLrUpmnmgmltgx+9OUkZFP1IAz+NAGnrmrtp0SQWyebfXJ2wRD19T7CnaJpA0u2YyN5t3Od88p6s3+FV9D0ycSvq2pc31wOF7Qp2UVt0AFFFFADX5RvpWH4M/5ACj0mkH/AI9W6ehrB8Gf8gMj0uJP50Ab9FFFABWdq+jw6tbhWYxTxndDMv3kNaNFAGJpGsTfaDpWqqIr6MfK38M6+o9626z9X0iDVrcI5Mc0Z3RTL96NvWqekavOtz/ZOrAR3yD5H/hnX1HvQBuUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAyVWeJ0VzGzKQHHVT61zuh+E7zQtiQ+ILmWAOZJI3t4szMepZtu4n3zXS0UAYNl4StbC5tryG5mF7HI7z3Jxuug/3lf1GcY9MDFbM1rb3BUzQRylful0Bx+dS0UAcvHo0l/wCPm1qazktoLKDykLsMXEnIDgAnhVJAJ5+b2roms7d7yO8aJTcRo0aSd1U4JH44H5VNRQBg6l4WGo3Vwx1W+htLwAXVpG42S4GOCQSoI4O0jNaradYvjdZwNtUKN0YOAOgqzSEhQSTgDqTQBh63otuSmpw6hNpE1rC0ZmtwuDETkqVYEdeRxkGs/wAMaKky2928bpZWoIsoZDljk5MjHuxJJ/Gpzu8V6jtGRpFq/J/5+HH9BXSgBVCqAABgAdqAFooooAKKKKACsDwb/wAgeUel1J/Ot+sDwdxplyPS8l/mKAN+iiigAooooAKoatpEGr2wjkykqHdFKv3o29RV+igDD0nV547r+ydXwl4o/dyfwzr6j3rcqhq2k2+rWvlS5SRTuilX70beoqjpWrXEN1/ZGr4S7Ufupf4Z19R70AbtFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVzuq3M2tXx0SwcrEv/AB+Tr/CP7o9zVjXNUmSRNL035r+4HB7RL3Y1d0nS4dJslt4vmb70kh6u3cmgCe1tYbO2jt7dAkcYwoFTUUUAFFFFABRRRQAVz/hD/jxvR6Xsv9K6Cuf8JcQaiPS+k/pQB0FFFFABRRRQAUUUUAFUdV0q31a18mbKspzHIv3o29RV6igDC0rVbi3uxpGsYW6H+pm/hnX/ABrdqlqulW+rWhgnBBHMci/eRvUVnaXqtxaXY0jWCBcD/UT/AMM4/wAaAN6iiigAooooAKKKKACiiigAooooAKKKKACs3WtXXSrUFV825mOyCIdXb/CrOoX8GmWUl1cNtRB07sewHvWVothPd3R1zU1xcSDEER6Qp2/E0AWND0hrCN7q7bzb+5O6aT0/2R7CtaiigAooooAKKKKACiiigArn/CnH9qL6Xz8V0Fc/4X4m1cel89AHQUUUUAFFFFABRRRQAUUUUAFUtU0u31a0NvcLgjlHH3kb1FXaKAMDTNUuLK7Gkaw377/lhcfwzj/Gt+qep6ZbataG3uF90cfeQ+orM0zU7nT7tdI1hv3v/Lvcn7sw9D70Ab9FFFABRRRQAUUUUAFFFFABTJZY4YmllYIiDLMegFZsniCESyrb2d3dpAxWWWCLcqkdR15I9s1RuxN4muobeEldI2LLJMp/1/oo/LmqcJJXaJUk3ZDbOKTxLqK6lcqV063b/RYm/wCWh/vkV0tNjjSKNY41CoowqjoBTqkoKKKKACiiigAooooAKKKKACuf8M8Xusj0vTXQVgeHONT1sf8AT5/SgDfooooAKKKKACiiigAooooAKKKKACqep6ZbaraNbXK5B5Vh1Q+oq5RQBz+m6nc6bdrpGsNlzxbXJ6Sj0PvXQVU1LTbbVbRra5TKnlWHVT6isnTtSudLu10jWHzni2uj0lHofegDoaKKr3zMmn3LKSrCJiCO3BppXYnoixRXARTk6TYrarrEeqXMa+RNLMwid8Aknc20jqcY5Fd6u7Yu7BbHOPWtatL2fUzp1OcdSVX1G/t9K0+e/u32QW6F3bGeBXnF18Wb0rK0WjLFER8rGbMiD+8Vxg+uM1MKU6l+VXsVKpCFuZ2ub8Opx6PoiabdXElnc28pEgWPc8ilydyD+IHPOMnrWv4U+bRzMMBZ7iWVUB+4GYkA+h9qXRG0+00uH/iZRXBkHmmV5QdxbnIyeBV4ahpsWdt3apk5OJFGT61UqiknpqyYwaa10RboqmdX00db+2/7+r/jTTrelDrqNt/39FYmpeorOOv6QOupW3/fwU0+I9FHXUrf/vugDTorKPifRB/zEoPzpD4p0Mf8xKL9f8KANaisc+LNDH/MQQ/RW/wpp8XaH/z+g/RG/wAKANqisQ+L9F7XLH6RN/hSf8Jfo/aSY/SFv8KANysDw9/yGNdH/T0P5Gnf8JfpXb7SfpA1Y+j+I7K31bVpWjuWWeZWULCScY7+lAHa0Vhf8JdY9rW+P0tzR/wllsfu6fqB/wC3c0AbtFYX/CUxn7ulaif+2FH/AAk7H7ui6kf+2P8A9egDdorC/wCEkuD93QdRP/bMUn/CRXp6eHr8/gBQBvUVg/8ACQaienh29/Flpf7d1Q9PDl1+Mi0AbtFYX9tayenh2b8Zlo/tfXT08PkfW4WgDdorC/tPxAemhIPrcCj+0PEh6aLAPrcCgDdqrqOnW2qWjW1ym5G6EdVPqPesz7b4mPTSbUfW4rmtR1HxImuSovmxS4X9zbkyJ0oA6HT9RudIu10nV33K3FtdnpIPQ+9aou7DUkntbe9glYoVdYpVZlzx0FeeeMZPFFz4Tf8AtC1UWglQzvtAkVM8nA6Dpk+ma4nTjLDqti+lYW+89Bb+T1bkZHHVcZz2xXVRw7qU5Tvaxz1a6hNQtue5y6JbzaHHpLs5SKNVjlHDqV+6wPqMVfjVkiRXcuwABYjG4+tPorncm9zdRS2KGt6VFrmi3emTOyJcxlN69VPY/gcV5enw08SyXX2aVrOOEnDXayk8eoTGc+x4969forSlXqUk1B7mdSlCpbmWxi2vhDQrS1ht10+NliQIGbJJwMZNTDw1oo/5hsH/AHzWpRWJqZo8PaOP+Ybb/wDfAp40LSR0062/79ir9FAFIaNpg6afbf8AfoU4aVpw6WNsP+2S/wCFW6KAKw06xHSytx/2yX/CnCxtB0tYR/2zFT0UARC1tx0t4h/wAUoghHSJB/wEVJRQAwRRjpGv5Uuxf7o/KnUUAJgDtWBofHiLXB/01Q/oa6Cuf0bjxPrY/wBqM/oaAOgooooAKKKKACiiigAooooAKKKKACiiigApMAEnHJ60tFACMoZSrAFSMEEdao2eh6Tp07XFlplpbSv954oVUn8QKv0UAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVz+k8eLdaHqIj+ldBXP6Zx4x1cescR/SgDoKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACsDT+PGuqj1giP6Vv1gWXHjfUh620Z/lQBv0UUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFYFpx45v/e0T+Yrfrn7fjx3d+9mv8xQB0FFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABXPxcePJveyH/oVdBXPjjx63vY/wDs1AHQUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFc+3Hj1PexP/oVdBWBNx48t/eyb/wBCoAdq2t3FlqSJCiNa2wR75iMlVdtq4+nLH2Fbtc/F4Xt7tbubVlMlxeSMZPLmcKF6KuAQDhcdq09Igu7XS4Le9dZJol2F1P3gOAfrjGa2qKHKuXdGUOe75updooorE1CiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAq6neHT9Ku70R+YbaB5Qg/i2qTj9K8ObxBrl3qCau2rTrelco0ZwiA87QnTb9ete9EBgQQCDwQe9cfL8LvDsl2ZR9rjgZsm1SbEf0HGQPYGurD1KVNv2kbnPXhUmlyOxueF9Vl1zw1YanNGI5biEM6r0z0JHscZFatRwQxW0EcEEaxxRqFRFGAoHAAqSuU6AooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAP/2Q==)
The point P lies on the y–axis, so we have to put x = 0 in the equation for PQ
i.e.2x –3y + 9 = 0
⇒ 2(0)–3y + 9 = 0
⇒ –3y + 9 = 0
⇒ –3y = –9
⇒ ![](data:image/png;base64,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)
Therefore, the point P is (0,3).
Now, to find the point Q which is on the x axis ,we have to substitute 0 in the place of y in the given equation QR.
2x –3y + 9 = 0
⇒ 2x–3(0) + 9 = 0
⇒ 2x + 9 = 0
⇒ 2x = –9
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAAgCAMAAABw3UvaAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOgA6OmY6OmZmOma2OpDbZgAAZgA6ZjoAZjpmZmYAZpDbZrbbZrb/kDoAkDo6kLbbkNv/tmYAtv//25A627Zm29u22////7Zm/9uQ/9u2//+2///bgkcWQAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA1ElEQVRIS9VUyxKCMAxM8IEvHgrVImhL+///aKoMyjhCG724px6yu2myLcAfQGcYHRh9mqQAHZfhTDU7A1S7cKJeSWjF0ptoEryjhBpxkzEcnZVJpbfja2HLM7RiXrD8viVViIsrDSp8o1UkwYpwHpHIcbAWaqIDaY5Ax+vt9IV7se7gGPW48EdRk+9dbp/wbNUepUn8M9vrW/dCFIaHVmBEjshgTg/0BxWXDNFjdW9OJj1Bw9wduJ+DhWE6/SUazhVdOpk8RTzFeIGPL5JB9B/FSOUNqkkJOj9oU64AAAAASUVORK5CYII=)
Therefore,the point Q becomes (![](data:image/png;base64,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)
Now to find the equation of PR, where P is (0,3) and Q is (
,we
have to use ,
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
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⇒ ![](data:image/png;base64,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)
⇒ –9y + 27 = 2(–3x)
⇒ –9y + 27 = –6x
⇒ –9y + 27 + 6x = 0
⇒ 2x–3y + 9 = 0(Divide by 3 on both the sides)
Find the slope of the straight line
3x + 4y – 6 = 0
Answer:
Here we have the straight line : 3x + 4y –6 = 0
The slope intercept form is y = mx + b, where m is the slope and b is the y intercept.
Therefore, 3x + 4y –6 = 0
⇒ 4y + 3x = 6
⇒ 4y = 6–3x
⇒ (Divide both sides of the equation by 4)
⇒
Now we will rewrite it in the slope intercept form
Hence according to the slope intercept form, y = mx + b
m i.e. slope is .
Question 2.
Find the slope of the straight line
y = 7x + 6
Answer:
Here we have the straight line : y = 7x + 6
The slope intercept form is y = mx + b, where m is the slope and b is the y intercept.
Therefore when, y = 7x + 6
Hence according to the slope intercept form,
y = mx + b
m i.e. slope is 7.
Question 3.
Find the slope of the straight line
4x = 5y + 3.
Answer:
Here we have the straight line : 4x = 5y + 3
The slope intercept form is y = mx + b, where m is the slope and b is the y intercept.
Therefore, 4x = 5y + 3
⇒ –5y – 3 = –4x (Commutative law)
⇒ 5y + 3 = 4x (multiply by – on both sides of the equation)
⇒ (Divide both sides of the equation by 5)
⇒
⇒
Now we will rewrite it in the slope intercept form
Hence according to the slope intercept form, y = mx + b
m i.e. slope is .
Question 4.
Show that the straight lines x + 2y + 1 = 0 and 3x + 6y + 2 = 0 are parallel.
Answer:
Given: Here the straight lines are x + 2y + 1 = 0 and
3x + 6y + 2 = 0.
To Prove: x + 2y + 1 = 0 and 3x + 6y + 2 = 0 are parallel.
Proof: If two lines are parallel then their slopes are equal.
Here slope of the first line x + 2y + 1 = 0 will be
(When the line is in the form ax + by + c = 0 then the slope of the line is )
⇒
Here slope of the second line 3x + 6y + 2 = 0 will be
⇒
⇒
Now both the slopes are equal.
Hence both the lines are parallel.
Question 5.
Show that the straight lines 3x – 5y + 7 = 0 and 15x + 9y + 4 = 0 are perpendicular.
Answer:
Given: The straight lines are 3x – 5y + 7 = 0 and
15x + 9y + 4 = 0 .
To Prove: The straight lines are 3x – 5y + 7 = 0 and
15x + 9y + 4 = 0 are perpendicular.
Proof: If two lines are perpendicular, then the product of their slopes is equal to –1.
The slope of the first line,3x –5y + 7 = 0 is
(
)
The slope of the second line, 15x + 9y + 4 = 0
is (
)
⇒
Now the product of these slopes is m1×m2.
⇒
As the product of the slopes is –1, the lines are perpendicular to
each other.
Question 6.
If the straight lines and ax + 5 = 3y are parallel, then find a.
Answer:
Given: The straight lines are and ax + 5 = 3y are parallel.
Since the lines are parallel their slopes should be equal.
Now,
⇒ y = 2(x–p)
⇒ y = 2x –2p
⇒ m1 = 2
ax + 5 = 3y
⇒ 3y = ax + 5
⇒
⇒
⇒
Now as the lines are parallel ,m1 = m2
⇒
⇒ 2×3 = a
⇒ a = 6
Question 7.
Find the value of a if the straight lines 5x – 2y – 9 = 0 and ay + 2x – 11 = 0 are perpendicular to each other.
Answer:
Given: The straight lines 5x – 2y – 9 = 0 and ay + 2x – 11 = 0 are perpendicular to each other.
As the lines are perpendicular to each other, the product of their slopes is equal to –1.
Slope of the first line 5x – 2y – 9 = 0 is m1.
(
)
Slope of the second line ay + 2x –11 = 0 is m2.
Therefore,
⇒
As the lines are perpendicular m1× m2 = –1
⇒
⇒ a = 5
Question 8.
Find the values of p for which the straight lines 8px + (2 – 3p)y + 1 = 0 and px + 8y – 7 = 0 are perpendicular to each other.
Answer:
Given: The straight lines 8px + (2 – 3p)y + 1 = 0 and
px + 8y – 7 = 0 are perpendicular to each other.
Since the lines are perpendicular to each other, product of their slopes is equal to –1.
Slope of the first line 8px + (2 – 3p)y + 1 = 0 is m1.
i.e.
Slope of the second line px + 8y – 7 = 0 is m2.
i.e.
As the lines are perpendicular to each other m1× m2 = –1.
⇒
⇒
⇒ p2 = –1×(2–3p)
⇒ p2 = 3p–2
⇒ p2–3p + 2 = 0
⇒ p.p–2p–p + 2 = 0
⇒ p(p–2)–1(p–2) = 0
⇒ (p–2)(p–1) = 0
⇒ p–2 = 0 ,p–1 = 0
⇒ p = 2,p = 1
Hence p = 1,2.
Question 9.
If the straight line passing through the points (h, 3)and (4, 1) intersects the line 7x – 9y – 19 = 0 at right angle, then find the value of h.
Answer:
Given: The straight line passing through the points (h, 3)
and (4, 1) intersects the line 7x – 9y – 19 = 0 at right angle.
As these lines intersect at right angle, they are perpendicular to each other.
When the lines are perpendicular to each other, the product of their slopes is equal to –1.
Slope of the first line passing through the points (h,3) and (4,1)
(here the two points are(x1,y1) and (x2,y2))
Now,
⇒
Slope of the second line 7x – 9y –19 = 0 is
Therefore, product of the slopes is
⇒
⇒
⇒
⇒ 14 = 36–9h
⇒ 9h = 36–14 = 22
⇒
Question 10.
Find the equation of the straight line parallel to the line 3x – y + 7 = 0 and passing through the point (1, –2).
Answer:
Here it’s given that the straight line is parallel to the line
3x – y + 7 = 0 and passing through the point (1, –2).
As the lines are parallel to each other their slopes are equal.
Slope of the given line 3x – y + 7 = 0 is
Equation of the line passing through the point(1,–2) is
(y–y1) = m(x–x1),where (x1,y1) is (1,–2)
⇒ (y–(–2)) = 3(x–1)
⇒ y + 2 = 3x–3
⇒ y + 2–3x + 3 = 0
⇒ –3x + y + 5 = 0
⇒ 3x–y–5 = 0 (multiplied by –1 on both sides of the equation)
Hence the equation of the line is 3x–y–5 = 0 .
Question 11.
Find the equation of the straight line perpendicular to the straight line x – 2y + 3 = 0 and passing through the point (1, –2).
Answer:
Here it’s given that the straight line is perpendicular to the
straight line x – 2y + 3 = 0 and passing through the point (1, –2).
As the lines are perpendicular to each other , the product of their
slopes is equal to –1.
Slope of the given line x–2y + 3 = 0 is
Equation of the line passing through the point(1,–2) is
(y–y1) = m(x–x1),where (x1,y1) is (1,–2)
)
⇒ 2(y + 2) = (x–1)
⇒ 2y + 4 = x–1
Question 12.
Find the equation of the perpendicular bisector of the straight line segment joining the points (3, 4) and (–1, 2).
Answer:
Given: There is a perpendicular bisector of the straight line segment joining the points (3, 4) and (–1, 2).
We have to find the equation of the perpendicular bisector.
As it is perpendicular to the given line segment,the product of their slopes is equal to –1 and as it bisects the line segment,it implies it divides the line segment into 2 equal parts.
Thus,mid–point of the line segment joining the points (3, 4) and (–1, 2) is:
; where (x1,y1) and (x2,y2) are the end points of the line segment
=
=
=
= 1,3
Therefore the mid–point is (1,3).
The slope of the line segment joining the points (3, 4) and
(–1, 2) is:
⇒
⇒
Therefore the equation of the perpendicular bisector is
(y–y1) = m(x–x1)
Now substitute the value of the mid–point(1,3) and slope in the above equation.
⇒ 2(y–3) = (x–1)
⇒ 2y –6 = x–1
⇒ 2y–6–x + 1 = 0
⇒ –x + 2y–5 = 0
⇒ x–2y + 5 = 0 (multiply by –1 on both the sides of the equation)
Question 13.
Find the equation of the straight line passing through the point of intersection of the lines 2x + y – 3 = 0 and 5x + y – 6 = 0 and parallel to the line joining the points (1, 2) and (2, 1).
Answer:
Here it is given that the straight line passing through the point of intersection of the lines 2x + y – 3 = 0 and 5x + y – 6 = 0 and parallel to the line joining the points (1, 2) and (2, 1).
As the straight line passes through the point of intersection of the lines 2x + y – 3 = 0 and 5x + y – 6 = 0 , we should find the intersection point by solving these equations:
i.e –3x = –3
⇒ x = 1
Substitute x = 1 in 2x + y–3 = 0
We get,2(1) + y–3 = 0
⇒ y–1 = 0
⇒ y = 1
Therefore , the intersection point is (1,1).
As the line passing through(1,1) is parallel to the line segments
joining the points (1, 2) and(2, 1),their slopes are equal.
Slope of the line joining the points (1, 2) and(2, 1) is
⇒
Hence the equation of the line passing through the point (1,1) with slope m equal to –1 is
(y–y1) = m(x–x1)
⇒ (y–1) = –1(x–1)
⇒ y–1 = –x + 1
⇒ y–1 + x–1 = 0
⇒ x + y–2 = 0
Question 14.
Find the equation of the straight line which passes through the point of intersection of the straight lines 5x – 6y = 1 and 3x + 2y + 5 = 0 and is perpendicular to the straight line 3x – 5y + 11 = 0.
Answer:
Here we have the straight line which passes through the point of intersection of the straight lines 5x – 6y = 1 and 3x + 2y + 5 = 0 and is perpendicular to the straight line 3x – 5y + 11 = 0.
As the straight line passes through the point of intersection of the lines 5x – 6y = 1 and 3x + 2y + 5 = 0, we should find the intersection point by solving these equations:
5x –6y = 1
3x + 2y + 5 = 0
⇒ 5x – 6y– 1 = 0 –––(1)
and 3x + 2y + 5 = 0–––(2)
Now multiply equation (2) by 3 on both the sides.
Thus we have 9x + 6y + 15 = 0–––(3)
Now, we have
⇒ 14x = –14
⇒ x = –1
Now, substituting x = –1 in the equation 5x–6y = 1, we have,
5(–1)–6y = 1
⇒ –5–6y = 1
⇒ –6y = 1 + 5 = 6
⇒
⇒ y = –1
Therefore the point of intersection is (–1,–1).
Slope of the line 3x –5y + 11 = 0 is :
As the line which passes through (–1,–1) is perpendicular to 3x –5y + 11 = 0 ,the product of their slopes will be –1.
Therefore,
⇒ (Here we have multiplied
on both the sides)
⇒
Hence the equation of the line passing through (–1,–1) and slope as is:
(y–y1) = m(x–x1)
⇒
⇒
⇒ 3y + 3 = –5x–5
⇒ 3y + 3 + 5x + 5 = 0
⇒ 5x + 3y + 8 = 0
Question 15.
Find the equation of the straight line joining the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4 and the point of intersection of the lines 2x + y – 4 = 0 and x – 2y + 3 = 0.
Answer:
Here we have the straight line which joins the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4 and the point of intersection of the lines 2x + y – 4 = 0 and x – 2y + 3 = 0.
As the straight line which joins the point of intersection of the lines 3x – y + 9 = 0 and x + 2y = 4,let us solve these 2 equations:
3x –y + 9 = 0–––(1)
X + 2y–4 = 0–––(2)
⇒ 3x –y + 9 = 0–––(1)
3x + 6y–12 = 0–––(2)(multiply (2) equation by 3 on both the sides)
Now,
(Subtract equation (2) from (1))
⇒ –7y = –21
⇒
Substitute y = 3 in the first equation
3x–y + 9 = 0
⇒ 3x–3 + 9 = 0
⇒ 3x + 6 = 0
⇒ 3x = –6
⇒
Thus, the point of intersection is(–2,3).
Now,
2x + y – 4 = 0–––(3)
x – 2y + 3 = 0–––(4)
(multiply (4) equation by 2 on both the sides) we get
2x – 4y + 6 = 0 --------(5)
Now,
⇒ 5y = 10
⇒
Substitute y = 2 in the (3) equation:
2x + y–4 = 0
⇒ 2x + 2–4 = 0
⇒ 2x–2 = 0
⇒ 2x = 2
⇒
The point of intersection is (1,2).
Hence equation of the line is
Here we have the points (–2,3) and (1,2)
Therefore,
⇒ 3(y–3) = –1(x + 2)
⇒ 3y–9 = –x–2
⇒ 3y–9 + x + 2 = 0
⇒ 3y + x–7 = 0
⇒ x + 3y–7 = 0
Question 16.
If the vertices of a Δ ABC are A(2, –4), B(3, 3) and C(–1, 5). Find the equation of the straight line along the altitude from the vertex B.
Answer:
Given: A ∆ABC has vertices A(2,–4),B(3,3) and C(–1,5).
The straight line BD drawn from vertex B is perpendicular to the line AC. So the product of the slopes of BD and AC is equal to –1.
Slope of AC is m1.
Where (x1,y1) and (x2,y2) are (2,–4) and (–1,5)
Therefore,
Slope of the BD is
=
Hence the equation of the line BD drawn from the vertex B is
(y–y1) = m(x–x1),here B is (3,3) and m =
⇒
⇒ 3(y–3) = (x–3)
⇒ 3y –9 = x–3
⇒ 3y–9–x + 3 = 0
⇒ –x + 3y–6 = 0
⇒ x–3y + 6 = 0
Question 17.
If the vertices of a Δ ABC are A(–4,4 ), B(8 ,4) and C(8,10). Find the equation of the straight line along the median from the vertex A.
Answer:
Given: The ∆ABC with vertices A(–4,4),B(8,4) and C(8,10) and
AD is the median,i.e. it passes through the mid point of BC.
Therefore D is midpoint of BC where B(8,4) and C(8,10).
=
=
=
Equation of AD is
,where (x1,y1) and (x2,y2) are (–4,4) and (8,7)
⇒
⇒
⇒
⇒ 12(y–4) = 3(x + 4)
⇒ 12y–48 = 3x + 12
⇒ 12y–48–3x–12 = 0
⇒ –3x + 12y–60 = 0
⇒ 3x–12y + 60 = 0
⇒ x–4y + 15 = 0(Divide both sides of the equation by 3)
Hence Equation of AD is x–4y + 15 = 0.
Question 18.
Find the coordinates of the foot of the perpendicular from the origin on the straight line 3x + 2y = 13.
Answer:
Here we have a perpendicular from the origin i.e.(0,0) to the
straight line 3x + 2y = 13.
We have to find the foot of the perpendicular i.e the intersection point at the line 3x + 2y = 13.
As these lines are perpendicular the product of their slopes is equal to –1.
Slope of the 3x + 2y–13 = 0 is m.
Therefore the slope of perpendicular is
i.e.
Hence the equation of the perpendicular from (0,0) and slope as
is (y–y1) = m(x–x1)
⇒
⇒ 3y = 2x
⇒ 3y–2x = 0
⇒ 2x–3y = 0
Now solve the two equations 3x + 2y–13 = 0 and 2x–3y = 0.
3x + 2y–13 = 0–––(1)
2x–3y = 0–––––(2)
Multiply (1) by 3 and (2) by 2 and add
⇒
Substitute x = 3 in the equation 2x–3y = 0.
2(3)–3y = 0
⇒ –3y = –6
⇒ y = 2(Divide both the sides of the equation by –3)
Hence the coordinates of the foot of the perpendicular is(3,2)
Question 19.
If x + 2y = 7 and 2x + y = 8 are the equations of the lines of two diameters of a circle, find the radius of the circle if the point (0, –2) lies on the circle.
Answer:
Given: The equations of the lines of two diameters of a circle
are x + 2y = 7 and 2x + y = 8 and F(0,–2)
Now to find the center of the circle,we have to find the
intersection of the lines x + 2y = 7 and 2x + y = 8
x + 2y = 7–––(1)
2x + y = 8–––(2)
Multiplying (1) by 2,we get
Substitute y = 2 in x + 2y = 7 we get,
x + 2(2) = 7
⇒ x = 7 – 4 = 3
The point of intersection is (3,2) i.e.the center.
Therefore distance between the points (3,2) and (0,–2) is
=
=
=
= 5 units
Hence the radius of the circle is 5 units.
Question 20.
Find the equation of the straight line segment whose end points are the point of intersection of the straight lines
2x – 3y + 4 = 0, x – 2y + 3 = 0 and the midpoint of the line joining the points (3, –2) and (–5, 8).
Answer:
Here its given that the straight line segment has end points as the point of intersection of the straight lines
2x – 3y + 4 = 0, x – 2y + 3 = 0 and the midpoint of the line joining the points (3, –2) and (–5, 8).
As one end point is point of intersection of the straight lines
2x – 3y + 4 = 0, x – 2y + 3 = 0,we will solve these equations.
2x–3y + 4 = 0––––(1)
x–2y + 3 = 0–––––(2)
Multiply (2) by 2,then we have
i.e. y = 2
Substitute y = 2 in 2x–3y + 4 = 0
2x–3(2) + 4 = 0
⇒ 2x–6 + 4 = 0
⇒ 2x–2 = 0
⇒ 2x = 2
⇒ x = 1
Therefore the lines intersect at (1,2).
Now the line has one end point as (1,2) and other end point as mid– point of the line joining (3, –2) and (–5, 8).
The mid– point of (3, –2) and (–5, 8) is
=
=
= (–1,3)
Hence to find the equation of the line with end points as (x1,y1) and (x2,y2) ,we use:
,where (x1,y1) and (x2,y2) are (1,2) and (–1,3)
⇒
⇒
⇒ –2(y–2) = (x–1)
⇒ –2y + 4 = x–1
⇒ –2y + 4–x + 1 = 0
⇒ –2y–x + 5 = 0
⇒ x + 2y–5 = 0(multiply by –1 on both the sides of the equation)
Question 21.
In an isosceles Δ PQR, PQ = PR. The base QR lies on the x–axis, P lies on the y– axis and 2x – 3y + 9 = 0 is the equation of PQ. Find the equation of the straight line along PR.
Answer:
Here it’s given that in an isosceles PQR, PQ = PR. The base QR lies on the x–axis, P lies on the y– axis and 2x – 3y + 9 = 0 is the equation of PQ.
The point P lies on the y–axis, so we have to put x = 0 in the equation for PQ
i.e.2x –3y + 9 = 0
⇒ 2(0)–3y + 9 = 0
⇒ –3y + 9 = 0
⇒ –3y = –9
⇒
Therefore, the point P is (0,3).
Now, to find the point Q which is on the x axis ,we have to substitute 0 in the place of y in the given equation QR.
2x –3y + 9 = 0
⇒ 2x–3(0) + 9 = 0
⇒ 2x + 9 = 0
⇒ 2x = –9
⇒
Therefore,the point Q becomes (
Now to find the equation of PR, where P is (0,3) and Q is (,we
have to use ,
⇒
⇒
⇒
⇒ –9y + 27 = 2(–3x)
⇒ –9y + 27 = –6x
⇒ –9y + 27 + 6x = 0
⇒ 2x–3y + 9 = 0(Divide by 3 on both the sides)
Exercise 5.6
Question 1.The midpoint of the line joining (a,– b) and (3a, 5b) is
A. (–a, 2b)
B. (2a, 4b)
C. (2a, 2b)
D. (–a,– 3b)
Answer:The midpoint of the line joining (a,– b) and (3a, 5b) is
,(mid point of line segment is
; where (x1,y1) and (x2,y2) are end points of the line segment)
i.e.![](data:image/png;base64,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)
= (2a,2b)
Question 2.The point P which divides the line segment joining the points A(1,– 3)and B(–3, 9) internally in the ratio 1:3 is
A. (2,1)
B. (0, 0)
C. ![](data:image/png;base64,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)
D. (1, –2)
Answer:Here P divides the line segment joining the points A(1,– 3)and B(–3, 9) internally in the ratio 1:3 .
Therefore coordinates of P are
;where k1 and k2 are the ratio in which the line is divided.
Now we substitute the values :
∴![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ xp = 0,yp = 0
Question 3.If the line segment joining the points A(3, 4) and B (14,– 3)meets the x–axis at P, then the ratio in which P divides the segment AB is
A. 4 : 3
B. 3 : 4
C. 2 : 3
D. 4 : 1
Answer:Here the line segment joining the points A(3, 4) and
B (14,– 3)meets the x–axis at P.
Therefore coordinates of P are
;where k1 and k2 are the ratio in which the line is divided.
Now,
![](data:image/png;base64,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)
⇒
,![](data:image/png;base64,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)
(here yp = 0∵ the line meets the x–axis at P)
⇒ ![](data:image/png;base64,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)
⇒ –3k1 + 4k2 = 0
⇒ –3k1 = –4k2
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADUAAAAhCAMAAABOfxPPAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADqQAGaQAGa2OgAAOgA6OjoAOmZmOma2OpDbZgAAZgA6ZjoAZjpmZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkNv/tmYAtmY6tv//25A625Bm27Zm27aQ29uQ29u229v/2////7Zm/9uQ/9u2//+2///b8hok2gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABBklEQVQ4T92U1w7CMAxFbWaZZZRdSJkpzf//H85iqCiJoOIBP1XqPbl2xgX4YTFsf+KWWkpsg/g8mpENM9qsNwqhxKJrqBTxmXd2zJK51opscpZK6+qieF9oCrF5UcIAqhjvxDxZSW0e6YECKLKgagLQznFsk5tIW9rTXapDjjiAlA5NLTPwMQDHqL7zq/5NQRfGVG1Tmu3+T394ZjfqqnfI2WHVZtWv92VunEaIfW9XWRdr08erKuI17N+c5+s6YmFU9BZNbkDeKd+CsrtSPXIDWMDzgusyecmNvX8soL4airK5wUIgAg4yRm1ucIK4DC1XFfEGZPja3CiG8j77KMgivfO/qxtkpRCJHF1wcgAAAABJRU5ErkJggg==)
Question 4.The centroid of the triangle with vertices at (–2,– 5), (–2,12) and (10, – 1)is
A. (6, 6)
B. (4, 4)
C. (3, 3)
D. (2, 2)
Answer:Given the triangle with vertices at (–2,– 5),
(–2,12) and (10, – 1).
The centroid of the triangle ABC is ![](data:image/png;base64,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)
∴ centroid of the given ∆ABC is![](data:image/png;base64,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)
i.e.
= (2,2)
Question 5.If (1, 2), (4, 6), (x, 6)and (3, 2)are the vertices of a parallelogram taken in order, then the value of x is
A. 6
B. 2
C. 1
D. 3
Answer:Here we have the parallelogram with vertices
(1, 2), (4, 6), (x, 6)and (3, 2).
Let the vertices be A(1,2),B(4,6),C(x,6) and D(3,2)
Since the vertices are taken in order AC and BD are the diagonals of the parallelogram.
In a parallelogram diagonals bisect each other.
∴Mid–point of AC = Mid–point of BD
Mid–point of two points (x1,y1) and (x2,y2) is
![](data:image/png;base64,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)
Here Mid–point of AC =
–––(1)
Mid–point of BD =
–––(2)
(1) = (2)
⇒ ![](data:image/png;base64,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)
Now we will equate the corresponding coordinates.
∴ ![](data:image/png;base64,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)
⇒ 2(1 + x) = 7×2
⇒ 2 + 2x = 14
⇒ 2x = 12
⇒ x = 6
Question 6.Area of the triangle formed by the points (0,0), (2, 0)and (0, 2)is
A. 1 sq. units
B. 2 sq. units
C. 4 sq. units
D. 8 sq. units
Answer:Here we have the triangle with vertices (0,0), (2, 0)and (0, 2).
Area of the ∆ABC with vertices as A(x1,y1),B(x2,y2) and C(x3,y3) is
![](data:image/png;base64,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)
Let us say the vertices are A(0,0),B(2,0) and C(0,2).
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAHUAAAAgCAMAAAAbmcfKAAAAAXNSR0IArs4c6QAAAJxQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmY6ZmZmZpCQZpC2ZpDbZrbbZrb/kDoAkLbbkNv/tmYAtmY6tpBmttuQttv/tv/btv//25A625Bm27Zm27aQ29u22/+22////7Zm/9uQ/9u2//+2///bKaYPMwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACAklEQVRIS+2Va3eCMAyGW92G29xFnLupTNwNdANG//9/W5K2XByXUjnHfbBf4JSkT/ImpIwdl6UCYn1m6bmHW3g1OwCVseBI3aNqJq5FhcXr+DP3Ca7fTQ7oYCO8ibQW/mmk/YR3m73jXnKxlDbKWHgnpe8deAqmDgo4LB2AjgRabLAi7IgemhpzuW29slwLJ8RDLW/iqOMDSk4b+0+OCtCSW0EVnv6HUvdRUWWyyji9+yCJY86XIcaVzPhgjt8355zfGERSQU0cWUXG/EmsqNJMGUPfBSRH4lwutqNV4izYFpwg3Ej4JuJXUDWJxeMoe/cxf81eAg8jS6TQPiSeCZTHTJ+wXWiVgmmgptMVy6koqTROxhoiCamLu4jGVabWqK2oKiAykupJBFAD0psOlcbU7hyNNFVviO29A7VuL2x9rtArcu1QUQLsJNgu5kpxnkU7uXZXGEOuqGtMUqYuFFpSs5KScReFi6IU/TRV1g6FUYzUBYkpYcx7wUQ4p+b6ecj+gAahG//XbDap4YTGLueQbAqP4RsUkZo4dPjgGRRYw9/67RoUtnk2QW1lx5dnU3u7tFhUUf/u7czh3qhfs8Isa7tz+qKm0xe2ycdHzf1aJYxVBIWDlIzdWs8KWnQK9ru97PgbkyvK7uh6r+AQ0BigscHc7jVZnDVGt0Wv1H9+2C+hzDjx8Cp6uAAAAABJRU5ErkJggg==)
⇒ Area = 2 sq.units
Hence Area is 2 sq.units.
Question 7.Area of the quadrilateral formed by the points (1,1), (0,1), (0, 0)and (1, 0)is
A. 3 sq. units
B. 2 sq. units
C. 4 sq. units
D. 1 sq. units
Answer:Here we have the quadrilateral formed by the points (1,1), (0,1), (0, 0)and (1, 0).
We have to calculate the Area of the quadrilateral with vertices as A(1,1), B(0,1), C(0, 0)and D(1, 0)
Let us divide the quadrilateral into 2 triangles,so Area of the quadrilateral will be sum of Areas of two triangles.
Let us say one ∆ is ABC and other ∆ is ADC
Now Area of ∆ABC is
![](data:image/png;base64,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)
When we substitute the values of the coordinates of the vertices
as A(1,1), B(0,1), C(0, 0),we get
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXEAAAAqCAMAAAByZ5sWAAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmY6ZpCQZpC2ZpDbZrbbZrb/kDoAkGY6kNv/tmYAtmY6tpBmttuQttv/tv/btv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///bpbi5ugAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEUUlEQVRoQ+2Z63rTMAyG4w1YBjtAYQO2dQswyChL2tz/zWHLJ8VSE7tN+3TPnD+wVv0sv5ZlWSmK/GQCmUASgcX1Q5J9Nt6OwPJKHP/ZTiL/OoVAe/FYZ+IpxCawzcQngJgkkYkn4ZrAOBOfAGKSRCaehGsC40x8AohJEpl4Eq4JjPvEl9e3E2i+dInm7HGHU6iP0J2zOc0XUMW6LXcVeN19KYR4c2NWtM3ALYnybodR7qS7+QkZpvt5Ht8FqC+Sd+Oh6tfvnidF3s0/MXoNTjD6+27+MWXk9kNiaBysfgtB7jh18y1XgCdeOVXbUPR2gy3G7l6IcwjvqLzkza2+/2RNYNnhN9QvipEWKdHX+93NvxE0GpP2AEt8NTNJxTcUG9PmGm4xqgBYzrUp3Y3dPIh7ZG70sQA3DzQ80SfyEpNzJ87/gtOHGs5xqr6VXFaIh84Sb42obyjaxD7SYoRs1AgAS4OQIPHmVh8LMJPAwxN9Spzqj/jP6oOK5bT68hvSSiPE3VMpv2mvxJGqN7q/74W4jADPEjfMIFJ1wOpkBs/QRalS1maHUGWCxJtbfSzAe++GJ/qUONUf8R9/7fTBN/tXfWIAtOXZ7eL0QVWPC2kgQ+a5q2ISDk/c/9JMEB2lA8T1qbKa6VOgCuudEAkyN/p9ARa5Hz7UZ5OWcSfO/35AWX3Y8YaTGkNHh8kD6sRzlR2KSzV/WW7rp7cQLHF0F9qAuD3N/fFr0PHEwbxHfLAcQMSDomENcaw/tkd7W8D6j4m35xawpruaqaTuLSMKNEPcLYcAp0iMo8BG/w1XUaNiiXMLjsyNaCDABQlPnI0nqh8EcZw+Jl7DT1TutMTtB93iWt4h44n3NzDdg3G70iYFnU4iYxzSfpBV6P3LOZgc41h/2xhffVa9DzjmcIzLz1Qex6ddYlahJ2cc8cIcVbp8Gs3jyNzq9wS2zeOMfsLJ6fxHJ2cDiQzW0F6MTHDABPp5nD/5fZ2Jv7fVofcvslap1eqb9dIpDj8k0Xpzq48FeI9djBN9WqtQ/QTiTt9Xh+aIXM1kWjGzbMRt0T3dwEG6/BrTgxm+AbmUjhotvRZjQEWVKTaNR9Tj3tzqY4E1xO0ZE1GPU/3+IcWN4Kbn9P0NaCaEDPKV/Of4l0zaEFFPpTj6LiP/h6zG/80iEvngLR81FO2dsN9iJC7LO5u40P2XiDunuuIZc3QntAIMD9zhjLhzUv2gRRoOwekHt/w1uSL+Y9fPWMjJv7U9cdrJ4ldmYJyovof//eHqB52seLQjebyS+Wip8hM8TLf2YHt7kQg29l/vpeSIWOuWVYLCAk4aeJgIPdT+dSTxYkP/zZk+PXHwG1WF+a2bCT3z1m064r0IgWrYPPnNMsTgTt8sxxXwsZs4240T2NHGGR/4tVpwb5NfK4v9zLuSrZj87JEAlJ3Ndq/w9ujuyx8KCvCuysT3tpSmpZ+J7414HigTODwC/wGCJ5+ubDJowgAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
Now Area of ∆ADC with vertices A(1,1), D(1, 0)
C(0, 0) is
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Hence area is 1 sq.units.
Question 8.The angle of inclination of a straight line parallel to x–axis is equal to
A. 0°
B. 60°
C. 45°
D. 90°
Answer:Since the line is parallel to x–axis makes an angle of
0 degree with x–axis,the angle of inclination becomes 0°.
Question 9.Slope of the line joining the points (3,– 2)and (–1, a) is
, then the value of a is equal to
A. 1
B. 2
C. 3
D. 4
Answer:Slope of the line joining the points (3,– 2)and (–1, a) is
.
Slope of the line with end points as (x1,y1) and (x2,y2)
is
.
∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJYAAAAhCAMAAAAMEKIdAAAAAXNSR0IArs4c6QAAAJlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOjqQOmZmOma2OpDbZgAAZgA6ZjoAZjpmZmaQZma2ZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bAeT6pgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACCUlEQVRYR+2XaVODMBCGk2qraK3Wth70sIcWWi0U/v+PcxMgbDiagxGdjvnAFJp998lm2SyE/I9WI7B//ij6818OrSJUOPMe0cOvCaVDuA/vbbkyhYbLkgCi8ZLsOiuQlGDVLuLZNJmUK6iN0IzjglIcnrL/8IZhhbdblS6WEljMKFEwGdHIJZse2iBJjyslcYrnKmlJSpIxjHTKH/QO0YjyMSVcL78lZMdSC7CyXTm5aJBiGyfU+ORUwSRaZHdH6aloeQmVDhaWQqvIFEywAshnr4vSphCVAKgClr3qTZSkchmhYIIVzWj3nV4jEykRku1gWOqUl6TyN1EomGCV59b4t0vbZiiSdWUu2JfT7L2CoPsOlGZ+sRk/c/iEzgDSmA6WxOP1+Y+M0IEUDR0ogvyXGOukNMFoTGsjlWJxNoxVGTWBqv6hivpJhRQmZ1OptfV/XbRsIl/DbCNltIltxQrOClaV05SXWpb2ECo8hQ4Urvzyqyxn7zze4HObEL9PO6+Wq9YoUJrK/sNEwornWTOvKYCmxfO+sm5qq3pytFhPY9yKJ848V6uj1SMrYR0Xrp5lYVYw1Gu0a8Wl3ptjoSdremmCJSyjp208c99sPy+LsOVN3KuP1ooVe/yMxJ8JVjFPjeL1FV5gNF7pnPjVHhtuIhLla8RnAzST1gXi07lQfvM2CeHZ2n4DGDctSBv0iSAAAAAASUVORK5CYII=)
Given
.
∴ ![](data:image/png;base64,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)
⇒ 2(a + 2) = –3(–4)
⇒ 2a + 4 = 12
⇒ a + 2 = 6 (Divide by 2 on both the sides)
⇒ a = 6–2 = 4
∴ a = 4
Question 10.Slope of the straight line which is perpendicular to the straight line joining the points (–2, 6)and (4, 8)is equal to
A. ![](data:image/png;base64,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)
B. 3
C. –3
D. ![](data:image/png;base64,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)
Answer:The straight line is perpendicular to the straight line joining the points(–2, 6)and (4, 8).
The slope of line joining the points (–2, 6)and (4, 8) is m1.
i.e. ![](data:image/png;base64,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)
Now the perpendicular line will have the slope m2 and as
the product of the slopes of 2 perpendicular lines is –1.
m1×m2 = –1
⇒ ![](data:image/png;base64,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)
⇒ m2 = –3
Question 11.The point of intersection of the straight lines 9x – y – 2 = 0 and 2x + y – 9 = 0 is
A. (–1, 7)
B. (7,1)
C. (1, 7)
D. (–1,– 7)
Answer:We can get the point of intersection of the straight lines 9x – y – 2 = 0 and 2x + y – 9 = 0 by solving these equations.
![](data:image/png;base64,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)
⇒ 11x = 11
⇒ x = 1
Substitute x = 1 in 9x–y–2 = 0.
⇒ 9(1)–y–2 = 0
⇒ 9–2–y = 0
⇒ 7 = y or y = 7
∴ The point of intersection is (1,7)
Question 12.The straight line 4x + 3y – 12 = 0 intersects the y– axis at
A. (3, 0)
B. (0, 4)
C. (3, 4)
D. (0, – 4)
Answer:When the straight line 4x + 3y – 12 = 0 intersects the y– axis ,then the x coordinate of that point is 0.Thus the point is(0,y).
Now we will substitute (0,y) in the equation for the straight line
4x + 3y – 12 = 0.
⇒ 4(0) + 3y–12 = 0
⇒ 3y–12 = 0
⇒ 3y = 12
⇒ ![](data:image/png;base64,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)
Hence the intersection point is (0,4).
Question 13.The slope of the straight line 7y – 2x = 11 is equal to
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:The slope of the line ax + by + c = 0 is
![](data:image/png;base64,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)
∴ slope of the straight line 7y – 2x = 11 is equal to
![](data:image/png;base64,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)
Question 14.The equation of a straight line passing through the point (2 , –7) and parallel to x–axis is
A. x = 2
B. x = – 7
C. y = – 7
D. y = 2
Answer:The equation of the straight line passing through the point (x1 , y1) with slope as m is
(y–y1) = m(x–x1)
So here the straight line passing through the point (2,–7) and parallel to x axis is
(y–(–7)) = 0(x–2) (Slope of the line parallel to x axis is 0)
⇒ y + 7 = 0
⇒ y = –7
Question 15.The x and y–intercepts of the line 2x – 3y + 6 = 0, respectively are
A. 2, 3
B. 3, 2
C. –3, 2
D. 3, –2
Answer:The x –intercept can be found by substituting y = 0 in the equation 2x–3y + 6 = 0.
⇒ 2x –3(0) + 6 = 0
⇒ 2x + 6 = 0
⇒ 2x = –6
⇒ ![](data:image/png;base64,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)
Now to calculate the y–intercept,substitute x = 0 in the equation 2x–3y + 6 = 0.
i.e.2(0)–3y + 6 = 0
⇒ –3y + 6 = 0
⇒ –3y = –6
⇒ ![](data:image/png;base64,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)
∴ x–intercept is –3 and y– intercept is 2 i.e.(–3,2).
Question 16.The centre of a circle is (–6, 4). If one end of the diameter of the circle is at (–12, 8), then the other end is at
A. (–18, 12)
B. (–9, 6)
C. (–3, 2)
D. (0, 0)
Answer:Here the centre of a circle is (–6, 4) and one end of the diameter of the circle is at (–12, 8).
As (–6,4) is the center of the circle it becomes the mid –point
of the diameter.
∴
is the mid–point or the center of the circle.
∴
)
⇒
and ![](data:image/png;base64,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)
⇒ x1–12 = –6×2 = –12
⇒ x1 = –12 + 12 = 0
Now, ![](data:image/png;base64,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)
⇒ y1 + 8 = 4×2
⇒ y1 = 8–8 = 0
Hence the other end of the diameter is (0,0).
Question 17.The equation of the straight line passing through the origin and perpendicular to the straight line 2x + 3y – 7 = 0 is
A. 2x + 3y = 0
B. 3x – 2y = 0
C. y + 5 = 0
D. y – 5 = 0
Answer:The straight line is passing through the origin and perpendicular to the straight line 2x + 3y – 7 = 0.
Since the straight line is perpendicular to the straight line
2x + 3y – 7 = 0,the product of their slopes will be equal to –1.
Slope of 2x + 3y–7 = 0 is m1.
![](data:image/png;base64,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)
∴ slope of the perpendicular line will be m2.
Now m1×m2 = –1
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
The equation of the line passing through (x1,y1) and slope as m is:
(y–y1) = m(x–x1)
∴ The equation of the line passing through the origin(0,0) and slope as
is:
![](data:image/png;base64,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)
⇒ 2(y–0) = 3(x–0)
⇒ 2y = 3x–0
⇒ 3x–2y = 0
Question 18.The equation of a straight line parallel to y–axis and passing through the point (–2, 5) is
A. x – 2 = 0
B. x + 2 = 0
C. y + 5 = 0
D. y – 5 = 0
Answer:The straight line parallel to y–axis and passing through the point (–2, 5) will have slope as tan 90°.
(∵ Slope of a line is tan θ, where θ is the angle formed by the line with the x–axis)
Equation of a line passing through (x1,y1) is
(y–y1) = m(x–x1),where m is the slope.
Here m = tan 90°.
i.e.
(y–5) = tan 90°(x–(–2))
⇒ ![](data:image/png;base64,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)
⇒ 0(y–5) = x + 2
⇒ 0 = x + 2
or x + 2 = 0
Question 19.If the points (2, 5), (4, 6) and (a, a) are collinear, then the value of a is equal to
A. –8
B. 4
C. –4
D. 8
Answer:Here we have the points (2, 5), (4, 6) and (a, a) collinear.
∴ Area of triangle formed by these vertices is 0.
Let us say that the vertices are A(2,5),B(4,6) and C(a,a).
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAcAAAAAqCAMAAAATU3sRAAAAAXNSR0IArs4c6QAAAKtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZmY6ZpCQZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLbbkNv/tmYAtmY6tpBmttuQttv/tv/btv//25A625Bm27Zm27aQ2/+22////7Zm/9uQ/9u2//+2///bGq+D0QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAFcUlEQVR4Xu1ZbWMbNQw+d7BlwGgZgW68rMsYPWAjJfQuuf//y/CLbMuW/JIeW3Ot71PT8+mRn0eSbbnr2tMYaAw8IAZ2r64f0Gwe3VT2l+LJ349u1g9nwuP5h20TcNl6NgGXrV/XBGwCLpyBhbvfMrAJuHAGFu5+y8Am4MIZWLj727PWiVmuhNP7lRDiizfLnUHzvJaB6c8XM3pu2/MPtUBk3Dzk7v6g5yAX2Jo237Mj0lxNm5e3d5ZAfjh+/faOn89FvkdomLQT0tE+bZ7OYrNLCEi5sncV8QfT7lKIL4ur5/TPV3bY+Lw42iuMzBNkb7IUEdb546C9fYalmrsb5Dwg2+h1Bgcxc+fBC0j+6+8qhqjl3Yurbr8u9sH7s9+7/cY4uyVBN21SWYnME2RkMqsgumgh0GnkDrkcQ3d1dzeYG0AGIR3B/S8rvgSWYtK+5wWMHfZ3FdPmWWi6V7+3olQW8TCaBxkBnfm7Inf4ooVA5wRMQ1fe3XCTNkJa2g8/ftQ1dBDi7c1Kxvd4Kc7UTlJXrIsKFVkBgateiKe3W2lap41JsnGlfk0b9Wpc2fwfeAHjYb2xQTFjGgNkNT0JyiODSbrfCTx0PQoCTQSMoLV9DjrZ9khwY5FNCNlf22dA7Lj65mr3/HpcXXU7SbFU4Hbqa6orK6BxuJNF5FqaMn+Dw4MxatZe+CEHJkpoMGz/3m5AdWTih9KIkME8h+xNMgoiDz3bMTTNQDxpsM9Bp/tWCW4A2RAOtCt0Q/ZoKmkvnXalxspg5iYjC55AV1ZAK4xyZQslOhTQ6CnjRz8hEubSDzushTiHDZdyMy8gRgbz1ineJBEw8BAJGEFTAT20c5lCu4CmuPCKcGMnrYUE2scXVi9D4WGt2LYj07QiVLDk1BXqncuscfXtBQwGDiwVyvj0zmwnfRCQKMHDZG2ARA0E5CMLIYN5DhmZJPGJoXkBEzGNoME+B41MxtAJbiB09OTtnPSnvkqrkDH/mHavZGdFiVp42Az0zvn2WpSBKs8H4xLZXQTx4YapwBAmncsZKMPYFgpr3pcFziSdJvIQ3VXH0NwmBvcUtcscdKb1z3PjMtALePhBZQBa4U0GGk5vo8J2lxLaHV7/ZFe3SEDFSW8CpJdYyccPk0Og0MsvovGURo9szbuywJpkHEDjjlkD8aTBZQ46d3fDcmMnjTLQZMBhLemw+yRgRiMeU0LD+buIe3etrasn2IVq2O9e6z2l3hcPqcMMDDOJDptVF2cOlQgoqzMgO/NuOpxJLoCch2jBItDMGgjQyOUYGvHBRi7HjUP2ayDUloM6RQM1gzxVTzdvdKjvf4a9ZDo9dK4y1IPDU6/KB5Q9qCu+Xvbmhd4W65H8Y4ZN6gy/h7ZR+Rzokb35GDkwyWKDhyrIbD0unwMdNLJPJo1NctAMNxbZCKlplwueDH617j35Sy54msIbeTT7VWbEH/IU+O+6YhHMnAPlgeZM5oGyjO4q3Bkf/oDSnBQQhu3VbceFqbXFTgxCRuZjZGySFRA+wBctxU4Mgkb2Y+jS3Q3DjUUOz4HZ5Kp6aQUMW5qkdeRt+S5evJDxeAMZdlRDEhk9Frm7P+gMctiJqdKoqoSGLU2+wWYMqT63XKLsGSJnnR027zaiElkdcaiHnwU6jxz3QmcraKWKWpq5mxt50/Rxc/FbRQJOzLA5V2P1yDLMqIefBzqLTG4j/jcBtSHU0szfncqVwnZVshlYN+yYSVQiq25tjYefADqHTO8Dj3GgPDbV0ix/2UacAgNVB8dTcLT5wDKQ27k0yk6fgVxL8/S9bx7mW5qNn5NnIN/SPHn3H72DpZbmoyfo1AkotTRP3f/mX2OgMdAYOIqB/wCmNexot4kCWAAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATwAAAAiCAMAAAD1cJVrAAAAAXNSR0IArs4c6QAAAJ9QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmZmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZmY6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kLaQkLbbkNv/tmYAtmY6tpBmttv/tv//25A625Bm27Zm27aQ29u229v/2/+22////7Zm/9uQ/9u2//+2///brDIPFAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADn0lEQVRoQ+1Y63qbMAzF69aVXbJkpcm2lvVC1tFkWyD4/Z9ttmxsYQtwYN/XJsM/esEHWTo+smSiaBoTA8/DAE8vnmfhF7nqgWwcCH+RIf87pw5kQ8P5ehKg2INB5D19Wk7kDSUvivKJvIm8ceffoLSdlKdIH08e/zH7OW4Dj+3tfP44hjyevd3pkHn6uf7z2EiQ/j49DPC6/HAzXHk5E+NSv69+R9F2BX7w7ZKxN60u9UyTkeSv+iNEdvktYzOtjQ5mylhGEWBaxLSOzxsKKd+pWEOuDNYdD16cqZzdL5n6K2PX0T7RT33fe6Z5qvYUjzIOiNDa5en5bp+6DviGy9hfiua6iBe/nZkcyAwhD7njwuv/y/ljrsmTTUzO2hzLuqcJ8qrkawh5xm4h0YXrwAjyivja31CQXgh5yB0XjrZPkwfreL43V4dpvnnP2MJxiyAvu4T1aTxhN5ObWCVOI9pFHumJsWxNIRxP5WkVQh5yx4WruGBg8uCN9iGnhaUdz1xN+TEWsx0sQuObayi7MqOqpHlIicduLph977FshNDAQQYFkIfdCSOv5zix0x7Qi7G6eojsDnUbhlnlrfqJBkXex9hWtnbL2dmdKIELba7GZWoVrO9MFiAYSBLYHYBriCM3qzylaRiUQTTdcLkVK8jLlW4Mvt0uQR4ZVcS/i8qW1nEaUlwGeMpEL7aNdUgd5JGZ5pGHUFTa9oi5nubblWgWnGRyBVLUsdyIHojAN7QFMqjzpPfM07shSOm0rDUslYZxhPI6yFPnZkjaZhedXbOelieIVVK9MNWqqLSl8cjhell9QhvxKwhpWBWWbsuaPJFXDRxx5tECR+4EVFtogQrHdxtkPQ2M9J958kVV7Um8b1e1SQGtSiZdLEU69lhWxU8oDeOqJLTaInfa+jxhXt8DoPnm4Bk1zLR0e//FbVRJgYBlGm+WsMvKQuvVC0J5stJDM91nORYpshE7gnHqihFQbaHua3fabhj8Vl52Xn8zJaKNPK1tMb0WJexP4hx6FHni3JPskXhDnrUrLzts7p4cvuH9SngMV/xuy1Ep75v3TVz4DQO545GHSmeL1k7x8SF3Wxu/Iu+XbHz0wyP/qjJoa8d8Vamu7qONaQSn73mhG2DSVgs39L3TxIUUDBS5geetvchp8kRGNZC8jfs55D+izK0AwaFrrvOJO0nZIOUVgrsi9Dts8MYcH3AIeVUir+sTeYcq7/jUMXl8Cgz8BfglgiSSMcsaAAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ –8 + a = 0
⇒ a = 8
Question 20.If a straight line y = 2x + k passes through the point (1, 2), then the value of k is equal to
A. 0
B. 4
C. 5
D. –3
Answer:When a straight line y = 2x + k passes through the point (1, 2), then the value of k can be obtained by substituting x = 1 and y = 2
in the equation of the straight line y = 2x + k.
i.e. 2 = 2×(1) + k
⇒ k + 2 = 2 (commutative property)
⇒ k = 2–2 = 0
Question 21.The equation of a straight line having slope 3 and y–intercept –4 is
A. 3x – y – 4 = 0
B. 3x + y – 4 = 0
C. 3x – y + 4 = 0
D. 3x + y + 4 = 0
Answer:The equation of a straight line having slope m and y–intercept as c is
y = mx + c
Now ,when slope is 3 and y–intercept –4 then equation of the straight line will be
y = 3x + (–4)
⇒ y = 3x–4
Question 22.The point of intersection of the straight lines y = 0 and x = – 4 is
A. (0,– 4)
B. (–4, 0)
C. (0, 4)
D. (4, 0)
Answer:The straight line x = –4 is a line parallel to y–axis and
perpendicular to x–axis and it intersects the x–axis or the straight
line y = 0 at (–4,0).
Question 23.The value of k if the straight lines 3x + 6y + 7 = 0 and 2x + ky = 5 are perpendicular is
A. 1
B. –1
C. 2
D. ![](data:image/png;base64,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)
Answer:When the straight lines 3x + 6y + 7 = 0 and 2x + ky = 5 are perpendicular the product of their slopes should be equal to –1 .
Slope of 3x + 6y + 7 = 0 is:
; (
)
Slope of 2x + ky = 5 is:
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFMAAAAqCAMAAAD4QZc7AAAAAXNSR0IArs4c6QAAAGBQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjpmOjqQOma2OpDbZgAAZjoAZrbbZrb/kDoAkLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2///bQum62AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABE0lEQVRIS+WW2xaCIBBFGSvJ7iZlF4j//8sGIl3pWqlxeijnwSfXlhkOW4QYXdnLhmhaQPtWtBW39eSIhKqUaSXtkEzP0l9gKmzvbplGwlu3+Qo9TZu7bcKWSq9YIOdoxkgN7d7M+QxZBWUq8gVlokf5njcmu2ii3UnSUvhHqFi7GJkVrL1sL8rkKelYu3iVGMmxqqTSsEsIHmev+mhHbgLTgx+i6meX5pceiXcVSDXYIaPt0lpnyy7xvSPs4nMT9shtFMAuRnJA64f42C5l71QMkI3um7Q/ZZZ8CzsM6K3rVTdPPi/QX6dj6gUUKZh5BiP5Skfk/vDI4nUaGa2f1xX5eYKvn56noIeJx5kUNv+hi8gdt4sQtmUL53QAAAAASUVORK5CYII=)
Now we have
.
∴ when we substitute the values of m1 and m2 we get:
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ 1 = (–1)×k
⇒ 1 = –k or k = –1
The midpoint of the line joining (a,– b) and (3a, 5b) is
A. (–a, 2b)
B. (2a, 4b)
C. (2a, 2b)
D. (–a,– 3b)
Answer:
The midpoint of the line joining (a,– b) and (3a, 5b) is
,(mid point of line segment is
; where (x1,y1) and (x2,y2) are end points of the line segment)
i.e.
= (2a,2b)
Question 2.
The point P which divides the line segment joining the points A(1,– 3)and B(–3, 9) internally in the ratio 1:3 is
A. (2,1)
B. (0, 0)
C.
D. (1, –2)
Answer:
Here P divides the line segment joining the points A(1,– 3)and B(–3, 9) internally in the ratio 1:3 .
Therefore coordinates of P are
;where k1 and k2 are the ratio in which the line is divided.
Now we substitute the values :
∴
⇒
⇒
⇒ xp = 0,yp = 0
Question 3.
If the line segment joining the points A(3, 4) and B (14,– 3)meets the x–axis at P, then the ratio in which P divides the segment AB is
A. 4 : 3
B. 3 : 4
C. 2 : 3
D. 4 : 1
Answer:
Here the line segment joining the points A(3, 4) and
B (14,– 3)meets the x–axis at P.
Therefore coordinates of P are
;where k1 and k2 are the ratio in which the line is divided.
Now,
⇒ ,
(here yp = 0∵ the line meets the x–axis at P)
⇒
⇒ –3k1 + 4k2 = 0
⇒ –3k1 = –4k2
⇒
Question 4.
The centroid of the triangle with vertices at (–2,– 5), (–2,12) and (10, – 1)is
A. (6, 6)
B. (4, 4)
C. (3, 3)
D. (2, 2)
Answer:
Given the triangle with vertices at (–2,– 5),
(–2,12) and (10, – 1).
The centroid of the triangle ABC is
∴ centroid of the given ∆ABC is
i.e. = (2,2)
Question 5.
If (1, 2), (4, 6), (x, 6)and (3, 2)are the vertices of a parallelogram taken in order, then the value of x is
A. 6
B. 2
C. 1
D. 3
Answer:
Here we have the parallelogram with vertices
(1, 2), (4, 6), (x, 6)and (3, 2).
Let the vertices be A(1,2),B(4,6),C(x,6) and D(3,2)
Since the vertices are taken in order AC and BD are the diagonals of the parallelogram.
In a parallelogram diagonals bisect each other.
∴Mid–point of AC = Mid–point of BD
Mid–point of two points (x1,y1) and (x2,y2) is
Here Mid–point of AC = –––(1)
Mid–point of BD = –––(2)
(1) = (2)
⇒
Now we will equate the corresponding coordinates.
∴
⇒ 2(1 + x) = 7×2
⇒ 2 + 2x = 14
⇒ 2x = 12
⇒ x = 6
Question 6.
Area of the triangle formed by the points (0,0), (2, 0)and (0, 2)is
A. 1 sq. units
B. 2 sq. units
C. 4 sq. units
D. 8 sq. units
Answer:
Here we have the triangle with vertices (0,0), (2, 0)and (0, 2).
Area of the ∆ABC with vertices as A(x1,y1),B(x2,y2) and C(x3,y3) is
Let us say the vertices are A(0,0),B(2,0) and C(0,2).
⇒
⇒
⇒
⇒ Area = 2 sq.units
Hence Area is 2 sq.units.
Question 7.
Area of the quadrilateral formed by the points (1,1), (0,1), (0, 0)and (1, 0)is
A. 3 sq. units
B. 2 sq. units
C. 4 sq. units
D. 1 sq. units
Answer:
Here we have the quadrilateral formed by the points (1,1), (0,1), (0, 0)and (1, 0).
We have to calculate the Area of the quadrilateral with vertices as A(1,1), B(0,1), C(0, 0)and D(1, 0)
Let us divide the quadrilateral into 2 triangles,so Area of the quadrilateral will be sum of Areas of two triangles.
Let us say one ∆ is ABC and other ∆ is ADC
Now Area of ∆ABC is
When we substitute the values of the coordinates of the vertices
as A(1,1), B(0,1), C(0, 0),we get
⇒
Now Area of ∆ADC with vertices A(1,1), D(1, 0)
C(0, 0) is
⇒
Hence area is 1 sq.units.
Question 8.
The angle of inclination of a straight line parallel to x–axis is equal to
A. 0°
B. 60°
C. 45°
D. 90°
Answer:
Since the line is parallel to x–axis makes an angle of
0 degree with x–axis,the angle of inclination becomes 0°.
Question 9.
Slope of the line joining the points (3,– 2)and (–1, a) is , then the value of a is equal to
A. 1
B. 2
C. 3
D. 4
Answer:
Slope of the line joining the points (3,– 2)and (–1, a) is .
Slope of the line with end points as (x1,y1) and (x2,y2)
is .
∴
Given .
∴
⇒ 2(a + 2) = –3(–4)
⇒ 2a + 4 = 12
⇒ a + 2 = 6 (Divide by 2 on both the sides)
⇒ a = 6–2 = 4
∴ a = 4
Question 10.
Slope of the straight line which is perpendicular to the straight line joining the points (–2, 6)and (4, 8)is equal to
A.
B. 3
C. –3
D.
Answer:
The straight line is perpendicular to the straight line joining the points(–2, 6)and (4, 8).
The slope of line joining the points (–2, 6)and (4, 8) is m1.
i.e.
Now the perpendicular line will have the slope m2 and as
the product of the slopes of 2 perpendicular lines is –1.
m1×m2 = –1
⇒
⇒ m2 = –3
Question 11.
The point of intersection of the straight lines 9x – y – 2 = 0 and 2x + y – 9 = 0 is
A. (–1, 7)
B. (7,1)
C. (1, 7)
D. (–1,– 7)
Answer:
We can get the point of intersection of the straight lines 9x – y – 2 = 0 and 2x + y – 9 = 0 by solving these equations.
⇒ 11x = 11
⇒ x = 1
Substitute x = 1 in 9x–y–2 = 0.
⇒ 9(1)–y–2 = 0
⇒ 9–2–y = 0
⇒ 7 = y or y = 7
∴ The point of intersection is (1,7)
Question 12.
The straight line 4x + 3y – 12 = 0 intersects the y– axis at
A. (3, 0)
B. (0, 4)
C. (3, 4)
D. (0, – 4)
Answer:
When the straight line 4x + 3y – 12 = 0 intersects the y– axis ,then the x coordinate of that point is 0.Thus the point is(0,y).
Now we will substitute (0,y) in the equation for the straight line
4x + 3y – 12 = 0.
⇒ 4(0) + 3y–12 = 0
⇒ 3y–12 = 0
⇒ 3y = 12
⇒
Hence the intersection point is (0,4).
Question 13.
The slope of the straight line 7y – 2x = 11 is equal to
A.
B.
C.
D.
Answer:
The slope of the line ax + by + c = 0 is
∴ slope of the straight line 7y – 2x = 11 is equal to
Question 14.
The equation of a straight line passing through the point (2 , –7) and parallel to x–axis is
A. x = 2
B. x = – 7
C. y = – 7
D. y = 2
Answer:
The equation of the straight line passing through the point (x1 , y1) with slope as m is
(y–y1) = m(x–x1)
So here the straight line passing through the point (2,–7) and parallel to x axis is
(y–(–7)) = 0(x–2) (Slope of the line parallel to x axis is 0)
⇒ y + 7 = 0
⇒ y = –7
Question 15.
The x and y–intercepts of the line 2x – 3y + 6 = 0, respectively are
A. 2, 3
B. 3, 2
C. –3, 2
D. 3, –2
Answer:
The x –intercept can be found by substituting y = 0 in the equation 2x–3y + 6 = 0.
⇒ 2x –3(0) + 6 = 0
⇒ 2x + 6 = 0
⇒ 2x = –6
⇒
Now to calculate the y–intercept,substitute x = 0 in the equation 2x–3y + 6 = 0.
i.e.2(0)–3y + 6 = 0
⇒ –3y + 6 = 0
⇒ –3y = –6
⇒
∴ x–intercept is –3 and y– intercept is 2 i.e.(–3,2).
Question 16.
The centre of a circle is (–6, 4). If one end of the diameter of the circle is at (–12, 8), then the other end is at
A. (–18, 12)
B. (–9, 6)
C. (–3, 2)
D. (0, 0)
Answer:
Here the centre of a circle is (–6, 4) and one end of the diameter of the circle is at (–12, 8).
As (–6,4) is the center of the circle it becomes the mid –point
of the diameter.
∴ is the mid–point or the center of the circle.
∴)
⇒ and
⇒ x1–12 = –6×2 = –12
⇒ x1 = –12 + 12 = 0
Now,
⇒ y1 + 8 = 4×2
⇒ y1 = 8–8 = 0
Hence the other end of the diameter is (0,0).
Question 17.
The equation of the straight line passing through the origin and perpendicular to the straight line 2x + 3y – 7 = 0 is
A. 2x + 3y = 0
B. 3x – 2y = 0
C. y + 5 = 0
D. y – 5 = 0
Answer:
The straight line is passing through the origin and perpendicular to the straight line 2x + 3y – 7 = 0.
Since the straight line is perpendicular to the straight line
2x + 3y – 7 = 0,the product of their slopes will be equal to –1.
Slope of 2x + 3y–7 = 0 is m1.
∴ slope of the perpendicular line will be m2.
Now m1×m2 = –1
⇒
⇒
The equation of the line passing through (x1,y1) and slope as m is:
(y–y1) = m(x–x1)
∴ The equation of the line passing through the origin(0,0) and slope as is:
⇒ 2(y–0) = 3(x–0)
⇒ 2y = 3x–0
⇒ 3x–2y = 0
Question 18.
The equation of a straight line parallel to y–axis and passing through the point (–2, 5) is
A. x – 2 = 0
B. x + 2 = 0
C. y + 5 = 0
D. y – 5 = 0
Answer:
The straight line parallel to y–axis and passing through the point (–2, 5) will have slope as tan 90°.
(∵ Slope of a line is tan θ, where θ is the angle formed by the line with the x–axis)
Equation of a line passing through (x1,y1) is
(y–y1) = m(x–x1),where m is the slope.
Here m = tan 90°.
i.e.
(y–5) = tan 90°(x–(–2))
⇒
⇒ 0(y–5) = x + 2
⇒ 0 = x + 2
or x + 2 = 0
Question 19.
If the points (2, 5), (4, 6) and (a, a) are collinear, then the value of a is equal to
A. –8
B. 4
C. –4
D. 8
Answer:
Here we have the points (2, 5), (4, 6) and (a, a) collinear.
∴ Area of triangle formed by these vertices is 0.
Let us say that the vertices are A(2,5),B(4,6) and C(a,a).
⇒
⇒
⇒
⇒
⇒ –8 + a = 0
⇒ a = 8
Question 20.
If a straight line y = 2x + k passes through the point (1, 2), then the value of k is equal to
A. 0
B. 4
C. 5
D. –3
Answer:
When a straight line y = 2x + k passes through the point (1, 2), then the value of k can be obtained by substituting x = 1 and y = 2
in the equation of the straight line y = 2x + k.
i.e. 2 = 2×(1) + k
⇒ k + 2 = 2 (commutative property)
⇒ k = 2–2 = 0
Question 21.
The equation of a straight line having slope 3 and y–intercept –4 is
A. 3x – y – 4 = 0
B. 3x + y – 4 = 0
C. 3x – y + 4 = 0
D. 3x + y + 4 = 0
Answer:
The equation of a straight line having slope m and y–intercept as c is
y = mx + c
Now ,when slope is 3 and y–intercept –4 then equation of the straight line will be
y = 3x + (–4)
⇒ y = 3x–4
Question 22.
The point of intersection of the straight lines y = 0 and x = – 4 is
A. (0,– 4)
B. (–4, 0)
C. (0, 4)
D. (4, 0)
Answer:
The straight line x = –4 is a line parallel to y–axis and
perpendicular to x–axis and it intersects the x–axis or the straight
line y = 0 at (–4,0).
Question 23.
The value of k if the straight lines 3x + 6y + 7 = 0 and 2x + ky = 5 are perpendicular is
A. 1
B. –1
C. 2
D.
Answer:
When the straight lines 3x + 6y + 7 = 0 and 2x + ky = 5 are perpendicular the product of their slopes should be equal to –1 .
Slope of 3x + 6y + 7 = 0 is:
; (
)
Slope of 2x + ky = 5 is:
Now we have .
∴ when we substitute the values of m1 and m2 we get:
⇒
⇒ 1 = (–1)×k
⇒ 1 = –k or k = –1