Question

Consider the line $L\equiv \frac{x-1}{2}=\frac{y+2}{3}=\frac{z-7}{6}.$ Point $P\left(2,-5,0\right)$ and $Q$ are such that $PQ$ is perpendicular to the line $L$ and the midpoint of $PQ$ lies on line $L,$ then coordinates of $Q$ are

• A. $\left(-4,-5,2\right)$
• B. $\left(-3,0,1\right)$
• C. $\left(1,6,2\right)$
• D. $\left(1,5,7\right)$

Answer 1

Answer: (A)

$Q$ is reflection of $P$ about line $L$

Coordinate of point $R$ are $(2\lambda +1,3\lambda -2,6\lambda +7)$ $\overrightarrow{PR}=⟨2\lambda -1,3\lambda +3,6\lambda +7⟩$
From diagram $\overrightarrow{PR}\cdot (2\hat{i}+3\hat{j}+6\hat{k})=0$
$4\lambda -2+9\lambda +9+36\lambda +42=0$
$49\lambda +49=0\Rightarrow \lambda =-1$
So, coordinates of $R$ are (-1,-5,1)
$\Rightarrow$ coordinates of $Q$ are (-4,-5,2)