Question

A line with positive direction cosines passes through the point $P(2,-1,2)$ and makes equal angles with the coordinate axes. The line meets the plane $2x+y+z=9$ at point $Q$. The length of the line segment $PQ$ equals

• A. $1$
• B. $\sqrt{2}$
• C. $\sqrt{3}$
• D. $2$

Answer 1

The correct option is C $\sqrt{3}$

Equation of the line is
$\frac{x-2}{1}=\frac{y+1}{1}=\frac{z-2}{1}=r\Rightarrow x=r+2,y=r-1,z=r+2$
The point $(r+2,r-1,r+2)$ lies on the plane $2x+y+z=9$ is $2(r+2)+r-1+r+2=9\Rightarrow r=1$
So, the coordinate of $Q$ are $(3,0,3)$ and thus
$PQ=\sqrt{{(3-2)}^2+{(0+1)}^2+{(3-2)}^2}=\sqrt{3}$