Question

A point $P$ lies on a line through $Q(1,-2,3)$ and is parallel to the line $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$. If $P$ lies on the plane $2x+3y-4z+22=0$, then segment $PQ$ equals to:

• A. $\sqrt{42}$
• B. $\sqrt{32}$
• C. $4$ units
• D. $5$ units

Answer 1

The correct option is A $\sqrt{42}$

Equation line $\frac{x-1}{1}=\frac{y+2}{4}=\frac{z-3}{5}=\lambda$
$P(\lambda +1,4\lambda -2,5\lambda +3)$
$P$ lies on $2x+3y-4z+22=0$
$2(\lambda +1)+3(4\lambda -2)-5(5\lambda +3)+22=0$
$\Rightarrow -6\lambda +6=0$
$\Rightarrow \lambda =1$
$P(2,2,8)$
$PQ=\sqrt{1+16+25}=\sqrt{42}$