Question

Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2x–y+z+3=0$ and let $R(3,5,\gamma )$ be a point of this plane. Then the square of the length of the line segment $SR$ is

Answer 1

Let $S$ be $(x_1,y_1,z_1)$
$\frac{x_1-1}{2}=\frac{y_1-3}{-1}=\frac{z_1-4}{1}=-2\cdot \frac{2\times 1-3+4}{2^2+(-1{)}^2+1^2}$
$\Rightarrow x_1=–3,y_1=5,z_1=2$
Hence, $S$ is $(–3,5,2)$
As $R(3,5,\gamma )$ lies on $2x–y+z+3=0$
$\Rightarrow 6–5+\gamma +3=0\Rightarrow \gamma =–4$
Hence, $R$ is $(3,5,–4)$
$\therefore (SR{)}^2=6^2+0^2+6^2=72$