Let S be the mirror image of the point Q(1,3,4) with respect to the plane 2x-y+z+3=0 and l…

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: 72

Accepted Answer

Foot of perpendicular from $Q$: $t=-\dfrac{2(1)-3+4+3}{6}=-1$, foot $=(-1,4,3)$, so $S=2(-1,4,3)-(1,3,4)=(-3,5,2)$.
$R$ on the plane: $6-5+\gamma+3=0\implies\gamma=-4$, so $R=(3,5,-4)$.
$$SR^2=(3+3)^2+(5-5)^2+(-4-2)^2=36+0+36=72.$$

Solutions · 1