Coordinate Geometry

1. Find the distance between P(b+c, c+a) and Q(c+a, a+b).
   
2. For what value of x, the distance between the points (x,2) and (3,4) be 8 units.
   
3. Find the point on x-axis which is equidistant from the points (-2,5) and (2,-3).
   
4. Find the point on y-axis which is equidistant from the points (-5,-2) and (3,2).
   
5. If the distances of the point P(x,y) from the points A(5,1) and B(-1,5) are equal, show that 3x=2y.
   
6. For what value of k, the point P(0,2) is equidistant from the points (3,k) and (k,5).
   
7. Prove that the points A(a,a), B(-a,-a) and C(-√3a,√3a) are the vertices of a an equilateral triangle. Calculate the area of this triangle.
   
8. Prove that A(-3,0), B(1,-3) and C(4,1) are the vertices of an isosceles right angled triangle. Find the area of this triangle.
   
9. Prove that the points A(1,-3), B(13,9), C(10,12) and D(-2,0) taken in order are the angular points of a rectangle.
   
10. Prove that the points A(1,2), B(5,4), C(3,8) and D(-1,6) taken in order are the angular points of a square.
    
11. Show that P(2,-1), Q(3,4), R(-2,3) and S(-3,-2) are four angular points of a rhombus but not a square. Also find its area.
    
12. Find the coordinates of the circumcentre of a triangle whose vertices are A(4,6), B(0,4), C(6,2). Also find its circumradius.
    
13. Prove that the points A(1,1), B(-2,7) and C(3,-3) are collinear.
    
14. Find the coordinates of the point which divides the line segment joining the points A(4, -3) and B(9,7) in the ratio 3:2.
    
15. Find the coordinates of the points which divide the line segment joining the points (-2,0) and (0,8) in four equal parts.
    
16. Find the ratio in which the point P(m,6) divides the join of A(-4,3) and B(2,8). Also find the value of m.
    
17. If A and B are the points (-2,-2) and (2,-4) respectively, find the coordinates of a point P on the line segment AB such that AP=(3/7)AB.
    
18. Find the coordinates of point A, where AB is a diameter of a circle whose centre is (2,-3) and B is the point (1,4).
    
19. Find the ratio in which line segment joining the points (1,-5) and (-4,5) is divided by x-axis. Also find the point of division.
    
20. Find the ratio in which line segment joining the points (1,3) and (2,7) is divided by the line 3x+y-9=0.
    
21. If the points A(6,1), B(8,2), C(9,4) and D(x,3) are vertices of a parallelogram taken in order. Find the value of x.
    
22. Find the value of p for which the points (-1,3),(2,p),(5,-1) are collinear.
    
23. Three consecutive vertices of a parallelogram are (-2,-1), (1,0) and (4,3). Find the coordinates of the fourth vertex.
    
24. Two vertices of a triangle are (1,2) and (3,5). If the centroid of the triangle is at origin, find the coordinates of the third vertex.
    

    
    Prove that the points (p,0), (0,q) and (1,1) are collinear if,  1  +  1  =  1
                                                                                                         p      q                                       

    
    
25. The coordinates of mid points of the sides of a triangle are (1,1), (2,-3) and (3,4) Find the coordinates of the vertices of triangle.
    
26. If the points (-1,3), (1,-1) and (5,1) are the vertices of a triangle, find the length of median through the third vertex.
    
27. The line segment joining the points (3,-4) and (1,2) is trisected at the points P(p,-2) and Q(5/3,q). Find the values of p and q.