Question

A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1 : n. Find the equation of the line.

Answer 1

Let point C divides the join of A(1, 0) and B(2, 3) in the ratio 1: n.

$\therefore$ Coordinates of C are $(\frac{2+n}{1+n},\frac{3}{1+n})$

Slope of $AB=\frac{3-0}{2-1}=3$

Since the required line is perpendicular to AB,

$\therefore$ Slope of required line $m=-\frac{1}{3}$

Now the required line passing through point $(\frac{2+n}{1+n},\frac{3}{1+n})$ having slope $-\frac{1}{3}$

$\therefore$ Equation of required line is

$y-\frac{3}{1+n}=\frac{-1}{3}(x-\frac{2+n}{1+n})$

$\Rightarrow \frac{(1+n)y-3}{1+n}=-\frac{1}{3}\left[\frac{(1+n)x-2-n}{1+n}\right]$

$\Rightarrow$ 3(1 + n)y - 9 = 1(1 + n)x + 2 + n

$\Rightarrow$ (1 + n) x + 3 (1 + n) y = n + 11.