Question

If the image of the point $P(1,-2,3)$ in the plane $2x+3y-$ $4z+22=0$ measured parallel to the line $20x=5y=4z$ is point $Q$, then the value of $|PQ{|}^2$ is

Answer 1

Answer: 168

Since, distance $PQ$ is measured parallel to the line $20x=5y=4z$
i.e. $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$
$\Rightarrow$ Direction ratios of line $PQ$ are $1,4,5$

Equation of $PQ$
$\Rightarrow \frac{x-1}{1}=\frac{y+2}{4}=\frac{z-3}{5}=\lambda$, (say)
$\Rightarrow$ A point on the line is $(\lambda +1,4\lambda -2,5\lambda +3)\equiv M$, (say)
Point $M$ lies on the plane $2x+3y-4z+22=0$
$\Rightarrow 2(\lambda +1)+3(4\lambda -2)-4(5\lambda +3)+22=0$
$\iff -6\lambda +6=0$
$\Rightarrow \lambda =1$
$\Rightarrow M=(2,2,8)$
$M$ is the midpoint of $PQ$
$\Rightarrow |PQ|=2|PM|$
$\Rightarrow |PQ|=2\sqrt{(2-1{)}^2+(2+2{)}^2+(8-3{)}^2}$
$=2\sqrt{1+16+25}$
$=2\sqrt{42}$
$\Rightarrow |PQ{|}^2=4\times 42=168$