The line joining the points A (-3, -10) and B (-2, 6) is divided by the point P such that $\dfrac{PB}{AB} = \dfrac{1}{5}$. Find the co-ordinates of P.

Let co-ordinates of P be (a, b).

Given,

$\dfrac{PB}{AB} = \dfrac{1}{5}$

Let PB = x and AB = 5x.

From figure,

⇒ AB = PA + PB

⇒ 5x = PA + x

⇒ PA = 4x.

$\dfrac{PA}{PB} = \dfrac{4x}{x} = \dfrac{4}{1}$.

PA : PB = 4 : 1.

∴ P divides the line segment joining A and B in ratio = 4 : 1.

By formula,

$x = \dfrac{m$1x2 + m2x1}{m1 + m2} \\[1em] = \dfrac{4 \times -2 + 1 \times -3}{4 + 1} \\[1em] = \dfrac{-8 - 3}{5} \\[1em] = -\dfrac{11}{5} \\[1em] y = \dfrac{m1y2 + m2y1}{m1 + m2} \\[1em] = \dfrac{4 \times 6 + 1 \times -10}{4 + 1} \\[1em] = \dfrac{24 - 10}{5} \\[1em] = \dfrac{14}{5}.

Hence, P = $\Big(-\dfrac{11}{5}, \dfrac{14}{5}\Big)$.