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.
Show SolutionHide Solution↓
(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 = $\frac{my_2 + ny_1}{m + n}$
Substituting values we get :
$$\begin{aligned}\Rightarrow 0 = \frac{m \times 10 + n \times -2}{m + n} \\[1em] \Rightarrow 0 = \frac{10m - 2n}{m + n} \\[1em] \Rightarrow 10m - 2n = 0 \\[1em] \Rightarrow 10m = 2n \\[1em] \Rightarrow \frac{m}{n} = \frac{2}{10} \\[1em] \Rightarrow \frac{m}{n} = \frac{1}{5} \\[1em] \Rightarrow m : n = 1 : 5.\end{aligned}$$
Hence, P divides AB in the ratio 1 : 5.
(b) Using section formula,
x = $\frac{mx_2 + nx_1}{m + n}$
Substituting values we get :
$$\begin{aligned}\Rightarrow x = \frac{1 \times 2 + 5 \times -10}{1 + 5} \\[1em] \Rightarrow x = \frac{2 - 50}{6} \\[1em] \Rightarrow x = \frac{-48}{6} \\[1em] \Rightarrow x = -8.\end{aligned}$$
P = (x, 0) = (-8, 0).
Hence, coordinates of point P are (−8, 0).