Question

M is the mid-point of the line segment joining the points A(–3, 7) and B(9, –1). Find the coordinates of point M. Further, if R(2, 2) divides the line segment joining M and the origin in the ratio p : q, find the ratio p : q.

Answer 1

Given, M is the mid-point of the line segment joining the points A(−3, 7) and B(9, −1).

The co-ordinates of point M are 

`((-3 + 9)/2, (7 - 1)/2)`

= `(6/2, 6/2)`

= (3, 3)

Also, given that, R(2, 2) divides the line segment joining M and the origin in the ratio p : q.

∴ `(2, 2) = ((p xx 0 + q xx 3)/(p + q),(p xx 0 + q xx 3)/(p + q))`

`=> (p xx 0 + q xx 3)/(p + q) = 2`

`=> (3q)/(p + q) = 2`

`=>` 3q = 2p + 2q

`=>` 3q – 2q = 2p

`=>` q = 2p

`=> p/q = 1/2`

Thus the ratio p : q is 1 : 2.