Question

If the image of the point $P(1,–2,3)$ in the plane, $2x+3y–4z+22=0$ measured parallel to the line,$\frac{x}{1}=\frac{y}{4}=\frac{z}{5}\text{is}Q,\text{then}PQ$ is equal to:

• A. $3\sqrt{5}$
• B. $2\sqrt{42}$
• C. $\sqrt{42}$
• D. $6\sqrt{5}$

Answer 1

The correct option is B $2\sqrt{42}$

Let midpoint of $PQ$ be $M$ which lies on the plane.
Equation of line $PQ$ is $(1,-2,3)+\lambda (1,4,5)$
$\Rightarrow \text{Let}M(x,y,z)=(1+\lambda ,4\lambda -2,5\lambda +3)\text{Putting this point in the equation of plane}2(1+\lambda )+3(4\lambda -2)-4(5\lambda +3)+22=0\Rightarrow \lambda =1\Rightarrow M(2,2,8),P(1,-2,3)PQ=2\times PM=2\sqrt{42}$