If A and B are (-2, -2) and (2, -4), respectively, find the coordinates of P such that AP = $\dfrac{3}{7}$ AB and P lies on the line segment AB.

Given,

AP = $\dfrac{3}{7}$ AB

From figure,

AB = AP + PB

PB = AB - AP = $AB - \dfrac{3}{7}AB = \dfrac{7AB - 3AB}{7} = \dfrac{4}{7}$ AB.

AP : PB = $\dfrac{3}{7}AB : \dfrac{4}{7}AB$ = 3 : 4.

∴ P divides line segment AB in the ratio 3 : 4.

Let co-ordinates of P be (x, y).

By section-formula,

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

Substituting values we get,

$\Rightarrow (x, y) = \Big(\dfrac{3 \times 2 + 4 \times -2}{3 + 4}, \dfrac{3 \times -4 + 4 \times -2}{3 + 4}\Big) \\[1em] \Rightarrow (x, y) = \Big(\dfrac{6 - 8}{7}, \dfrac{-12 - 8}{7}\Big) \\[1em] \Rightarrow (x, y) = \Big(-\dfrac{2}{7}, -\dfrac{20}{7}\Big).$

Hence, P = $\Big(-\dfrac{2}{7}, -\dfrac{20}{7}\Big).$