Q. 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} $ is $Q$, then $PQ$ is equal to :
Solution:
Equation of PQ, $\frac{x-1}{1}=\frac{y+ 2}{4}=\frac{z-3}{5}$
Let M be $\left(\lambda+1,4\lambda-2,5\lambda+3\right)$
As it lies on $2x + 3y - 4z + 22 = 0$
$\lambda=1$
For $Q, \lambda = 2$
Distance $PQ=2\sqrt{1^{2}+4^{2}+5^{2}}=2\sqrt{42}$
