Given that `A_1,A_2,A_3, A_n` are `n` points in a plane whose coordinates are `x_1,y_1),(x_2,y_2) ,(x_n ,y_n),` respectively. `A_1A_2` is bisected at the point `P_1,P_1A_3` is divided in the ratio `A :2` at `P_2,P_2A_4` is divided in the ratio 1:3 at `P_3,P_3A_5` is divided in the ratio `1:4` at `P_4` , and so on until all `n` points are exhausted. Find the final point so obtained.

Let A_1,A_2,A_3,...,A_n are n Points in a plane whose coordinates are (x_1,y_1),(x_2,y_2),....,(x_n,y_n) respectively. A_1A_2 is bisected at the point P_1,P_1A_3 is divided in the ratio 1:2 at P_2,P_2A_4 is divided in the ratio 1:3 at P_3,P_3A_5 is divided in the ratio 1:4 at P_4 and the so on until all n points are exhausted. find the coordinates of the final point so obtained.

If A_(1),A_(2),A_(3),…,A_(n) are n points in a plane whose coordinates are (x_(1),y_(1)),(x_(2),y_(2)),(x_(3),y_(3)),…,(x_(n),y_(n)) respectively. A_(1)A_(2) is bisected in the point G_(1) : G_(1)A_(3) is divided at G_(2) in the ratio 1 : 2, G_(3)A_(5) at G_(4) in the1 : 4 and so on untill all the points are exhausted. Show that the coordinates of the final point so obtained are (x_(1)+x_(2)+.....+ x_(n))/(n) and (y_(1)+y_(2)+.....+ y_(n))/(n)

A(x_(1),y_(1)), B(x_(2),y_(2)), C(x_(3),y_(3)) are three vertices of a triangle ABC. lx +my +n = 0 is an equation of the line L. If P divides BC in the ratio 2:1 and Q divides CA in the ratio 1:3 then R divides AB in the ratio (P,Q,R are the points as in problem 1)

The line segment joining the points A(3,2) and B (5,1) is divided at the point P in the ratio 1 : 2 and it lies on the line 3x-18y+k=0 . Find the value of k.

If point P(3,2) divides the line segment AB internally in the ratio of 3:2 and point Q(-2,3) divides AB externally in the ratio 4:3 then find the coordinates of points A and B.

Find the ratio in which point P(2,1) divides the line joining the points A (4,2) and B(8,4)

If x_1,x_2,x_3 as well as y_1, y_2, y_3 are in G.P. with same common ratio, then prove that the points (x_1, y_1),(x_2,y_2),a n d(x_3, y_3) are collinear.

If x_1,x_2,x_3 as well as y_1, y_2, y_3 are in G.P. with same common ratio, then prove that the points (x_1, y_1),(x_2,y_2),a n d(x_3, y_3) are collinear.

If the point P(-1,2) divides the join of points A(2, 5) and B(a, b) in the ratio 3 : 4, find the value of a xx b - a .

Let P be a point on x^2 = 4y . The segment joining A (0,-1) and P is divided by point Q in the ratio 1:2, then locus of point Q is