A(−10, −2) and B(2, 10) are two end points of a line segment. If AB intersects the x-axis at P, find the :

(a) ratio in which ‘P’ divides AB.

(b) coordinates of point P.

(a) Let point at which AB intersects x-axis be P(x, 0) and P divides AB in the ratio m : n.

Using section formula,

y = $\dfrac{my$2 + ny1}{m + n}

Substituting values we get :

$\Rightarrow 0 = \dfrac{m \times 10 + n \times -2}{m + n} \\[1em] \Rightarrow 0 = \dfrac{10m - 2n}{m + n} \\[1em] \Rightarrow 10m - 2n = 0 \\[1em] \Rightarrow 10m = 2n \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{2}{10} \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{1}{5} \\[1em] \Rightarrow m : n = 1 : 5.$

Hence, P divides AB in the ratio 1 : 5.

(b) Using section formula,

x = $\dfrac{mx$2 + nx1}{m + n}

Substituting values we get :

$\Rightarrow x = \dfrac{1 \times 2 + 5 \times -10}{1 + 5} \\[1em] \Rightarrow x = \dfrac{2 - 50}{6} \\[1em] \Rightarrow x = \dfrac{-48}{6} \\[1em] \Rightarrow x = -8.$

P = (x, 0) = (-8, 0).

Hence, coordinates of point P are (−8, 0).