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 the image of A (2, 1) be B (5, 2). Also, let M be the midpoint of AB.

$$\begin{gathered} \therefore \mathrm{Coordinates}\mathrm{of}M=\left(\frac{2+5}{2},\frac{1+2}{2}\right) \\ =\left(\frac{7}{2},\frac{3}{2}\right) \end{gathered}$$


Let CD be the mirror.
Line AB is perpendicular to the mirror CD.

$\therefore$ Slope of AB $\times$ Slope of CD = −1

$$\begin{gathered} \Rightarrow \frac{2-1}{5-2}\times \mathrm{Slope}\mathrm{of}CD=-1 \\ \Rightarrow \mathrm{Slope}\mathrm{of}CD=-3 \end{gathered}$$

Equation of the mirror CD:

$$\begin{gathered} y-\frac{3}{2}=-3\left(x-\frac{7}{2}\right) \\ \Rightarrow 2y-3=-6x+21 \\ \Rightarrow 6x+2y-24=0 \\ \Rightarrow 3x+y-12=0 \end{gathered}$$