Calculate the ratio in which the line joining the points (-3, -1) and (5, 7) is divided by the line x = 2. Also, find the co-ordinates of the point of intersection.

Let point of intersection be (2, y). [∵ any point on the line x = 2 has x co-ordinate = 2]

By formula,

$x = \dfrac{m$1x2 + m2x1}{m1 + m2} \\[1em] \Rightarrow 2 = \dfrac{m1 \times 5 + m2 \times -3}{m1 + m2} \\[1em] \Rightarrow 2m1 + 2m2 = 5m1 - 3m2 \\[1em] \Rightarrow 5m1 - 2m1 = 2m2 + 3m2 \\[1em] \Rightarrow 3m1 = 5m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{5}{3}.

m₁ : m₂ = 5 : 3.

$y = \dfrac{m$1y2 + m2y1}{m1 + m2} \\[1em] = \dfrac{5 \times 7 + 3 \times -1}{5 + 3} \\[1em] = \dfrac{35 - 3}{8} \\[1em] = \dfrac{32}{8} \\[1em] = 4.

Hence, co-ordinates of point of intersection = (2, 4) and ratio = 5 : 3.