Question

If the image of the point $(2,1)$ with respect to the line mirror be $(5,2),$ find the equation of the mirror.

Answer 1

Let $Q(5,2)$ be the mirror image of $P(2,-1)$ with respect to the line mirror $AB\times (ax+by+c=0)$

Then,

(Slope of AB) $\times$ (Slope of PQ) = - 1

$\frac{-a}{b}\times \left(\frac{2-1}{5-2}\right)=-1$

$\frac{-a}{b}\times \frac{1}{3}=-1$

$-a=-3b$

$a=3b...\left(1\right)$

and

(R) mid point of PQ should line in AB as PQ perpendicularly biosects AB

$\therefore$ Coordinates of R are

$(\frac{5+2}{2},\frac{2+1}{2})=(\frac{7}{2},\frac{3}{2})$

$\therefore \frac{7}{2}a+\frac{3}{2}b+c=0$

$7a+3\left(\frac{a}{3}\right)+2c=0[\because b=\frac{a}{3}from(1\left)\right]$

$8a+2c=0$

or, $-4a=6...\left(2\right)$

$\therefore$ equation of line is $ax+by+c=0$

or, $ax+\frac{a}{3}y-4a=0$

or, $3x+y-12=0$