Question

The mirror image of the point $(1,2,3)$ in a plane is $(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3})$. Which of the following points lies on this plane?

• A. $(1,-1,1)$
• B. $(-1,-1,1)$
• C. $(1,1,1)$
• D. $(-1,-1,-1)$

Answer 1

The correct option is A $(1,-1,1)$

Image of point $P(1,2,3)$ w.r.t. a plane $ax+by+cz+d=0$ is
$Q(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3})$

Direction ratios of $PQ:(-\frac{7}{3}-1,-\frac{4}{3}-2,-\frac{1}{3}-3)=(-\frac{10}{3},-\frac{10}{3},-\frac{10}{3})=(1,1,1)$
Direction ratios of normal to plane is $(1,1,1)$
Mid-point of line $PQ$ lies on the plane.

$\therefore$ The mid-point of $PQ=(-\frac{2}{3},\frac{1}{3},\frac{4}{3})$
$\therefore$Equation of plane is $x+\frac{2}{3}+y-\frac{1}{3}+z-\frac{4}{3}=0$
$\Rightarrow x+y+z=1$
Hence point $(1,-1,1)$ satisfies the equation of the plane.