Co-ordinate Geometry 1. Find the value of P for which the points (1, -), (2, -6) and (p, -1) are collinear 2. If the point P (x, y) is equidistant from the points A (1,) and B(4, 1). Prove that 2x+y = 7. If the points P(-1, y) and Q(5, 7) are on the circle with center A(2, ). Find the value of y. 4. In what ratio does the line x+2y = 18 divides the line segment joining (1, 5) and (-1, 10) 5. Show that the points A(-4, -1), B(2, -1) and C(2, 7) are the vertices of a right triangle. 6. Find the point on the y-axis that is equidistant from (-5,-2) and (,2). 7. Find the value of k, if the point (0,2) is equidistant from (,k) and (k,5). 8. If the distance of P(x,y) from A(5,1) and B(-1,5) are equal, prove that x = 2y 9. Find the value of k for which the points with co-ordinates (;2), (4,k) and (5,) are collinear. 10. Find the value of k for which the points with co-ordinates (;2), (4,k) and (1,5) are collinear. 11. Find the ratio in which the point (11, 15) divides the line segment joining the points (15,5) and (9,20). 12. The co-ordinates of the centroid of a triangle are (1,) and two of its vertices are (-7,6) and (8,5). Find the third vertex of the triangle.
1. Show that the points A(1,0), B(5,), C(2,7) and D(-2.-4) are the vertices of a parallelogram 14. The co-ordinates of the mid-point of the line joining the points (p,4) and (-2,2q) are (5,p). Find the value of p and q. 15. Find the ratio in which the line segment joining the pints (6,4) and (1,-7) is divided by the x-axis. 16. The centroid of a triangle is (1,4) and two of its vertices are (-9,7) and (8,1). Find the third vertex of the triangle. 17. Show that the points (1, -1), (5, 2) and (9, 5) are collinear 18. Find the value of x such that PQ = QR where the coordinates of P, Q and R are (6, -1), (1,) and (x, 8) respectively. 19. Find a point on X axis which is equidistant from the points (7, 6) and (-, 4) 20. If the points (n, 4), (5, 2) and (7, n) are collinear, find the value of n. 21. A (1,7) is the center of a circle with radius 5cm. If P(x, ) and Q (5, x) are the points on its circumference find the value of x. 22. Find a point on the Y-axis which is equidistant from the points A(-4,) & B(5,-6). 2. A point P is at a distance of 5 from the point A(4, 6). Find the coordinates of the point P if its y coordinate is three greater than its x coordinate. 24. If the point P(x, y) is equidistant from the points A (-2, 1) and B (,-1). Prove that x = 1 2 + 2 5 y. 25. The coordinates of the mid point of the line joining the points (p, 4) and (-2, 2q) is (5, p). Find the value of p and q. 26. Two points P(m, n) and Q(5, -1) on a line parallel to X - axis are equidistant from the point A (1, ), find the value of m and n.
27. If the points A(4, ) and B(x, 5) are on the circle with the center O(2,), find the value of x. 28. Find the ratio in which the point (x, -1) divides the line segment joining the points (-, 5) and (2, -5). Also find the value of x. 29. Find the area of the quadrilateral ABCD whose vertices are A (1,0), B(5,), C(2,7) and D(-2,4) 0. Find the area of the triangle formed by joining the mid points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, ) 1. Determine the ratio in which the line x+y 9 = 0 divides the line segment joining the points (1, ) and (2, 7). 2. If the distance of P(x, y) from the points A (, 6) and B(-, 4) are equal. Prove that x+y = 5.. If P divides the join of A(-2, -2) and B(2, -4) such that AP AB =, find the coordinates 7 of P. 4. The mid-points of the sides of a triangle are (,4), (4, 6) and (5,7). Find the coordinates of the vertices of the triangle. 5. If A (5, -1), B (-, -2) and C(-1, 8) are the vertices of triangle ABC, find the length of median through A and the coordinates of the centroid. 6. Show that the points (7, 10), (-2, 5) and (, -4) are the vertices of an isosceles right triangle. 7. If the point C(-1, 2) divides the line segment AB in the ratio :4, where the coordinates of A is (2,5), find the coordinates of B. 8. Find the ratio in which C(p, 1) divides the join of A(-4, 4) and B(6, -1) and hence find the value of p.
9. Show that the points A(1, 2), B(5, 4), C(,8) and D(-1, 6) are the vertices of a square. 40. Find the co-ordinates of the point equidistant from three given points A (5,1), B(-,-7) and C(7,-1) 41. The line-segment joining the points (, -4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, -2) and ( 5, q) respectively, find the values of p and q. 42. The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q. If point P lies on the line 2x y + k = 0, find the value of k. 4. Determine the ratio in which the point P(m, 6) divides the join of A(-4, ) and B(2,8). Also find the value of m. 44. Which point one of y axis is equidistant from (2,) and (-4,1) 45. Three vartices A, B and C of a parallelogram are (-2,-1), (1,0) and (-4,). Find the 4th vertex D. 46 Find the ratio in which the line segment joining (-2, -) and (5,6) is divided by x axis. Also find the co-ordinates of the point of intersection of the segment and x axis. 47. Show that the points A(2,-2), B(14,10), C(11,1) and D(-1,1) are the vertices of a rectangle. 48. Prove that the co-ordinates of the centroid of a ABC, with vertices A (x 1, y 2 ), B(x 2, y 2 ) and C(x, y ) are given by ( x 1 +x 2 + x, y 1 +y 2 + y 49. In what ratio does the point (-,7) divide the join of P(-5,11) and Q(4,-7) 50. Three vertices of a parallelogram are (a+b)(a b), (2a+b, 2a b), (a b, a+b). Find the fourth vertex. )
51. Show that the points (8,2), (0,0) and (5,) forms an isosceles right triangle. 52. Find a point on y-axis which is equidistant from the points A(-,4) and B(7,6) 5. In which ratio the point (-,p) divides the line segment joining the points (-5,-4) and (-2,). Hence find the value of p. 54. Find the point on y-axis which is equidistant from(-5,-2) and (,2).