Class 10 Exam > Class 10 Notes > Mathematics (Maths) Class 10 > RD Sharma Solutions: Coordinate Geometry (Exercise 7B)
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Page 1 1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio 2 : 3 Sol: The end points of AB are 1,7 4, 3 . A and B Therefore, 1 1 2 2 1 , 7 4, 3 x y and x y Also, 2 3 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 2 4 3 1 2 3 3 7 , 2 3 2 3 8 3 6 21 , 5 5 5 15 , 5 5 mx nx my ny x y m n m n x y x y x y Therefore, 1 3 x and y Hence, the coordinates of the required point are 1,3 . 2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the ratio 7 : 2 Sol: The end points of AB are 5,11 4, 7 . A and B Therefore, 1 1 2 2 5, 11 4, 7 x y and x y Also, 7 2 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 7 4 2 5 7 7 2 11 , 7 2 7 2 28 10 49 22 , 9 9 mx nx my ny x y m n m n x y x y Page 2 1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio 2 : 3 Sol: The end points of AB are 1,7 4, 3 . A and B Therefore, 1 1 2 2 1 , 7 4, 3 x y and x y Also, 2 3 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 2 4 3 1 2 3 3 7 , 2 3 2 3 8 3 6 21 , 5 5 5 15 , 5 5 mx nx my ny x y m n m n x y x y x y Therefore, 1 3 x and y Hence, the coordinates of the required point are 1,3 . 2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the ratio 7 : 2 Sol: The end points of AB are 5,11 4, 7 . A and B Therefore, 1 1 2 2 5, 11 4, 7 x y and x y Also, 7 2 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 7 4 2 5 7 7 2 11 , 7 2 7 2 28 10 49 22 , 9 9 mx nx my ny x y m n m n x y x y 18 27 , 9 9 x y Therefore, 2 3 x and y Hence, the required point are 2, 3 . P 3. If the coordinates of points A and B are (-2, -2) and (2, -4) respectively. Find the coordinates of the point P such that AP= 3 7 AB, where P lies on the segment AB. Sol: The coordinates of the points A and Bare 2, 2 and 2, 4 respectively, where 3 7 AP AB and P lies on the line segment AB. So 7 3 3 7 AP BP AB AP AP BP AP AB 7 3 3 4 AP BP AP AP BP Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally Then 3 2 4 2 6 8 2 3 4 7 7 3 4 4 2 12 8 20 3 4 7 7 x y Hence, the coordinates of point Pare 2 20 , . 7 7 4. Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that 2 5 PA PQ . If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k. Sol: Let the coordinates of A be , . x y Here 2 . 5 PA PQ So, PA AQ PQ 5 2 2 5 PA PA AQ PA PQ Page 3 1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio 2 : 3 Sol: The end points of AB are 1,7 4, 3 . A and B Therefore, 1 1 2 2 1 , 7 4, 3 x y and x y Also, 2 3 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 2 4 3 1 2 3 3 7 , 2 3 2 3 8 3 6 21 , 5 5 5 15 , 5 5 mx nx my ny x y m n m n x y x y x y Therefore, 1 3 x and y Hence, the coordinates of the required point are 1,3 . 2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the ratio 7 : 2 Sol: The end points of AB are 5,11 4, 7 . A and B Therefore, 1 1 2 2 5, 11 4, 7 x y and x y Also, 7 2 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 7 4 2 5 7 7 2 11 , 7 2 7 2 28 10 49 22 , 9 9 mx nx my ny x y m n m n x y x y 18 27 , 9 9 x y Therefore, 2 3 x and y Hence, the required point are 2, 3 . P 3. If the coordinates of points A and B are (-2, -2) and (2, -4) respectively. Find the coordinates of the point P such that AP= 3 7 AB, where P lies on the segment AB. Sol: The coordinates of the points A and Bare 2, 2 and 2, 4 respectively, where 3 7 AP AB and P lies on the line segment AB. So 7 3 3 7 AP BP AB AP AP BP AP AB 7 3 3 4 AP BP AP AP BP Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally Then 3 2 4 2 6 8 2 3 4 7 7 3 4 4 2 12 8 20 3 4 7 7 x y Hence, the coordinates of point Pare 2 20 , . 7 7 4. Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that 2 5 PA PQ . If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k. Sol: Let the coordinates of A be , . x y Here 2 . 5 PA PQ So, PA AQ PQ 5 2 2 5 PA PA AQ PA PQ 5 2 PA AQ PA 3 2 AQ PA 2 3 PA AQ Let (x, y) be the coordinates of A, which dives PQ in the ratio 2 : 3 internally Then using section formula, we get 2 4 3 6 8 18 10 2 2 3 5 5 2 1 3 6 2 18 20 4 2 3 5 5 x y Now, the point 2, 4 lies on the line3 1 0, x k y therefore 3 2 4 1 0 3 6 6 2 3 k k k Hence, 2. k 5. Points P, Q, R and S divide the line segment joining the points A(1,2) and B(6,7) in five equal parts. Find the coordinates of the points P,Q and R Sol: Since, the points P, Q, R and S divide the line segment joining the points 1,2 6,7 A and B in five equal parts, so AP PQ QR R SB Here, point P divides AB in the ratio of 1 : 4 internally So using section formula, we get Coordinates of 1 6 4 1 1 7 4 2 , 1 4 1 4 P 6 4 7 8 , 2,3 5 5 The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get Coordinates of 2 6 3 1 2 7 3 2 , 2 3 2 3 Q 12 3 14 6 , 3,4 5 5 The point R divides AB in the ratio of 3 : 2 internally So using section formula, we get Page 4 1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio 2 : 3 Sol: The end points of AB are 1,7 4, 3 . A and B Therefore, 1 1 2 2 1 , 7 4, 3 x y and x y Also, 2 3 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 2 4 3 1 2 3 3 7 , 2 3 2 3 8 3 6 21 , 5 5 5 15 , 5 5 mx nx my ny x y m n m n x y x y x y Therefore, 1 3 x and y Hence, the coordinates of the required point are 1,3 . 2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the ratio 7 : 2 Sol: The end points of AB are 5,11 4, 7 . A and B Therefore, 1 1 2 2 5, 11 4, 7 x y and x y Also, 7 2 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 7 4 2 5 7 7 2 11 , 7 2 7 2 28 10 49 22 , 9 9 mx nx my ny x y m n m n x y x y 18 27 , 9 9 x y Therefore, 2 3 x and y Hence, the required point are 2, 3 . P 3. If the coordinates of points A and B are (-2, -2) and (2, -4) respectively. Find the coordinates of the point P such that AP= 3 7 AB, where P lies on the segment AB. Sol: The coordinates of the points A and Bare 2, 2 and 2, 4 respectively, where 3 7 AP AB and P lies on the line segment AB. So 7 3 3 7 AP BP AB AP AP BP AP AB 7 3 3 4 AP BP AP AP BP Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally Then 3 2 4 2 6 8 2 3 4 7 7 3 4 4 2 12 8 20 3 4 7 7 x y Hence, the coordinates of point Pare 2 20 , . 7 7 4. Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that 2 5 PA PQ . If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k. Sol: Let the coordinates of A be , . x y Here 2 . 5 PA PQ So, PA AQ PQ 5 2 2 5 PA PA AQ PA PQ 5 2 PA AQ PA 3 2 AQ PA 2 3 PA AQ Let (x, y) be the coordinates of A, which dives PQ in the ratio 2 : 3 internally Then using section formula, we get 2 4 3 6 8 18 10 2 2 3 5 5 2 1 3 6 2 18 20 4 2 3 5 5 x y Now, the point 2, 4 lies on the line3 1 0, x k y therefore 3 2 4 1 0 3 6 6 2 3 k k k Hence, 2. k 5. Points P, Q, R and S divide the line segment joining the points A(1,2) and B(6,7) in five equal parts. Find the coordinates of the points P,Q and R Sol: Since, the points P, Q, R and S divide the line segment joining the points 1,2 6,7 A and B in five equal parts, so AP PQ QR R SB Here, point P divides AB in the ratio of 1 : 4 internally So using section formula, we get Coordinates of 1 6 4 1 1 7 4 2 , 1 4 1 4 P 6 4 7 8 , 2,3 5 5 The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get Coordinates of 2 6 3 1 2 7 3 2 , 2 3 2 3 Q 12 3 14 6 , 3,4 5 5 The point R divides AB in the ratio of 3 : 2 internally So using section formula, we get Coordinates of 3 6 2 1 3 7 2 2 , 3 2 3 2 R 18 2 21 4 , 4,5 5 5 Hence, the coordinates of the points P, Q and R are 2,3 , 3,4 and 4,5 respectively 6. Points P, Q, and R in that order are dividing line segment joining A (1,6) and B(5, -2) in four equal parts. Find the coordinates of P, Q and R. Sol: The given points are 1,6 A and 5, 2 . B Then, , P x y is a point that devices the line AB in the ratio 1: 3 By the section formula: 2 1 2 1 , 1 2 3 6 1 5 3 1 , 1 3 1 3 5 3 2 18 , 4 4 8 16 , 4 4 2 4 mx nx my ny x y m n m n x y x y x y x and y Therefore, the coordinates of point P are 2,4 Let Q be the mid-point of AB Then, , Q x y 1 2 1 2 , 2 2 6 2 1 5 , 2 2 6 4 , 2 2 3, 2 x x y y x y x y x y x y Therefore, the coordinates of Q are 3,2 Let , R x y be a point that divides AB in the ratio 3 :1 Then, by the section formula: Page 5 1. Find the coordinates of the point which divides the join of 1,7 4, 3 . A and B in the ratio 2 : 3 Sol: The end points of AB are 1,7 4, 3 . A and B Therefore, 1 1 2 2 1 , 7 4, 3 x y and x y Also, 2 3 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 2 4 3 1 2 3 3 7 , 2 3 2 3 8 3 6 21 , 5 5 5 15 , 5 5 mx nx my ny x y m n m n x y x y x y Therefore, 1 3 x and y Hence, the coordinates of the required point are 1,3 . 2. Find the co-ordinates of the point which divides the join of A(-5, 11) and B(4,-7) in the ratio 7 : 2 Sol: The end points of AB are 5,11 4, 7 . A and B Therefore, 1 1 2 2 5, 11 4, 7 x y and x y Also, 7 2 m and n Let the required point be , . P x y By section formula, we get 2 1 2 1 , 7 4 2 5 7 7 2 11 , 7 2 7 2 28 10 49 22 , 9 9 mx nx my ny x y m n m n x y x y 18 27 , 9 9 x y Therefore, 2 3 x and y Hence, the required point are 2, 3 . P 3. If the coordinates of points A and B are (-2, -2) and (2, -4) respectively. Find the coordinates of the point P such that AP= 3 7 AB, where P lies on the segment AB. Sol: The coordinates of the points A and Bare 2, 2 and 2, 4 respectively, where 3 7 AP AB and P lies on the line segment AB. So 7 3 3 7 AP BP AB AP AP BP AP AB 7 3 3 4 AP BP AP AP BP Let (x, y) be the coordinates of P which divides AB in the ratio 3 : 4 internally Then 3 2 4 2 6 8 2 3 4 7 7 3 4 4 2 12 8 20 3 4 7 7 x y Hence, the coordinates of point Pare 2 20 , . 7 7 4. Point A lies on the line segment PQ joining P(6, -6) and Q(-4, -1) in such a way that 2 5 PA PQ . If that point A also lies on the line 3x + k( y + 1 ) = 0, find the value of k. Sol: Let the coordinates of A be , . x y Here 2 . 5 PA PQ So, PA AQ PQ 5 2 2 5 PA PA AQ PA PQ 5 2 PA AQ PA 3 2 AQ PA 2 3 PA AQ Let (x, y) be the coordinates of A, which dives PQ in the ratio 2 : 3 internally Then using section formula, we get 2 4 3 6 8 18 10 2 2 3 5 5 2 1 3 6 2 18 20 4 2 3 5 5 x y Now, the point 2, 4 lies on the line3 1 0, x k y therefore 3 2 4 1 0 3 6 6 2 3 k k k Hence, 2. k 5. Points P, Q, R and S divide the line segment joining the points A(1,2) and B(6,7) in five equal parts. Find the coordinates of the points P,Q and R Sol: Since, the points P, Q, R and S divide the line segment joining the points 1,2 6,7 A and B in five equal parts, so AP PQ QR R SB Here, point P divides AB in the ratio of 1 : 4 internally So using section formula, we get Coordinates of 1 6 4 1 1 7 4 2 , 1 4 1 4 P 6 4 7 8 , 2,3 5 5 The point Q divides AB in the ratio of 2 : 3 internally. So using section formula, we get Coordinates of 2 6 3 1 2 7 3 2 , 2 3 2 3 Q 12 3 14 6 , 3,4 5 5 The point R divides AB in the ratio of 3 : 2 internally So using section formula, we get Coordinates of 3 6 2 1 3 7 2 2 , 3 2 3 2 R 18 2 21 4 , 4,5 5 5 Hence, the coordinates of the points P, Q and R are 2,3 , 3,4 and 4,5 respectively 6. Points P, Q, and R in that order are dividing line segment joining A (1,6) and B(5, -2) in four equal parts. Find the coordinates of P, Q and R. Sol: The given points are 1,6 A and 5, 2 . B Then, , P x y is a point that devices the line AB in the ratio 1: 3 By the section formula: 2 1 2 1 , 1 2 3 6 1 5 3 1 , 1 3 1 3 5 3 2 18 , 4 4 8 16 , 4 4 2 4 mx nx my ny x y m n m n x y x y x y x and y Therefore, the coordinates of point P are 2,4 Let Q be the mid-point of AB Then, , Q x y 1 2 1 2 , 2 2 6 2 1 5 , 2 2 6 4 , 2 2 3, 2 x x y y x y x y x y x y Therefore, the coordinates of Q are 3,2 Let , R x y be a point that divides AB in the ratio 3 :1 Then, by the section formula: 2 1 2 1 , mx nx my ny x y m n m n 3 2 1 6 3 5 1 1 , 3 1 3 1 15 1 6 6 , 4 4 16 0 , 4 4 4 0 x y x y x y x and y Therefore, the coordinates of R are 4,0 . Hence, the coordinates of point P, Q and R are 2,4 , 3,2 4,0 and respectively. 7. The line segment joining the points A(3,-4) and B(1,2) is trisected at the points P(p, -2) and 5 , 3 Q q . Find the values of p and q. Sol: Let P and Q be the points of trisection of AB. Then, P divides AB in the radio 1:2 So, the coordinates of P are 2 1 2 1 , 1 1 2 3 1 2 2 4 , 1 2 1 2 1 6 2 8 , 3 3 7 6 , 3 3 7 , 2 3 mx nx my ny x y m n m n x y x y x y x y Hence, the coordinates of P are 7 , 2 3 But , 2 p are the coordinates of P. So, 7 3 p Also, Q divides the line AB in the ratio 2:1 So, the coordinates of Q areRead More
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