If the image of the point P(1,-2,3) in the plane 2x+3y-4z+22=0 measured parallel to line x…

Question

If the image of the point $P(1,-2,3)$ in the plane $2x+3y-4z+22=0$ measured parallel to line $\dfrac x1=\dfrac y4=\dfrac z5$ is $Q$, then $PQ$ is equal to:

Accepted Answer

Let $F=(\lambda+1,4\lambda-2,5\lambda+3)$ lie on the plane:
$$2(\lambda+1)+3(4\lambda-2)-4(5\lambda+3)+22=0\implies -6\lambda+6=0\implies\lambda=1,\quad F=(2,2,8).$$
$\overrightarrow{PF}=(1,4,5)$, $|PF|=\sqrt{42}$. Since $F$ is the midpoint of $PQ$: $PQ=2PF=2\sqrt{42}$.

Solutions · 1