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

By formula,

Mid-point (M) = $\Big(\dfrac{x$1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

Substituting values we get,

$M = \Big(\dfrac{-3 + 9}{2}, \dfrac{7 + (-1)}{2}\Big) \\[1em] = \Big(\dfrac{6}{2}, \dfrac{6}{2}\Big) \\[1em] = (3, 3).$

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

By section formula,

$\Rightarrow x = \dfrac{m$1x2 + m2x1}{m1 + m2}

Substituting values we get,

$\Rightarrow 2 = \dfrac{p \times 0 + q \times 3}{p + q} \\[1em] \Rightarrow 2(p + q) = 0 + 3q \\[1em] \Rightarrow 2p + 2q = 3q \\[1em] \Rightarrow 2p = q \\[1em] \Rightarrow \dfrac{p}{q} = \dfrac{1}{2}.$

p : q = 1 : 2.

Hence, M = (3, 3) and p : q = 1 : 2.