Question

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

Answer 1

Let $A=(2,1)$ & $A^{{}^{\prime }}=(5,2)$

$\therefore$The line is mirror.

$\therefore$ Line is perpendicular to $A{A}^{{}^{\prime }}$ and passes through the midpoint.

Slope of $A{A}^{{}^{\prime }}=\frac{1}{3}$

$\therefore$ Slope of mirror$=-3$

$\therefore$ Equation $=(y-\frac{3}{2})=-3(x-\frac{7}{2})$

$\Rightarrow y-\frac{3}{2}=-3x+\frac{21}{2}$

$\Rightarrow 2y-3=-6x+21$

$\Rightarrow 6x+2y-24=0$

$\Rightarrow 3x+y-12=0$