Question

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

$2\sqrt{42}$

Any line parallel to $\frac{x}{1}=\frac{y}{4}=\frac{z}{5}$ and passing through P(1, -2, 3) is

$\frac{x-1}{1}=\frac{y+2}{4}=\frac{z-3}{5}=\lambda \left(say\right)$
Any point on above line can be written as
$(\lambda +1,4\lambda -2,5\lambda +3).\therefore$ Coordinates of R are $(\lambda +1,4\lambda -2,5\lambda +3)$Since, point R lies on the above plane.
$\therefore 2(\lambda +1)+3(4\lambda -2)-4(5\lambda +3)+22=0\Rightarrow \lambda =1$
So, point R is (2, 2, 8).
Now, $PR=\sqrt{(2-1{)}^2+(2+2{)}^2+(8-3{)}^2}=\sqrt{42}PQ=2PR=2\sqrt{42}$