The line segment joining the points A(4, -3) and B(4, 2) is divided by the point P such that AP : AB = 2 : 5. Find the co-ordinates of P.

Let point P be (x, y).

$\Rightarrow \dfrac{AP}{AB} = \dfrac{2}{5} \\[1em] \Rightarrow \dfrac{AP}{AP + PB} = \dfrac{2}{5} \\[1em] \Rightarrow 5AP = 2AP + 2PB \\[1em] \Rightarrow 3AP = 2PB \\[1em] \Rightarrow \dfrac{AP}{PB} = \dfrac{2}{3} \\[1em] \Rightarrow AP : PB = 2 : 3.$

m₁ : m₂ = AP : PB = 2 : 3.

By section-formula,

(x, y) = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

Substituting values we get :

$\Rightarrow (x, y) = \Big(\dfrac{2 \times 4+ 3 \times 4}{2 + 3}, \dfrac{2 \times 2 + 3 \times -3}{2 + 3}\Big) \\[1em] \Rightarrow (x, y) = \Big(\dfrac{8 + 12}{5}, \dfrac{4 - 9}{5}\Big) \\[1em] \Rightarrow (x, y) = \Big(\dfrac{20}{5}, \dfrac{-5}{5}\Big) \\[1em] \Rightarrow (x, y) = (4, -1).$

Hence, the coordinates of P are (4, -1).