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  4. Consider the line L≡ (x - 1/2)=(y + 2/3)=(z - 7/6). Point P(2 , - 5,0) 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

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Q. 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

NTA AbhyasNTA Abhyas 2020

Solution:

$Q$ is reflection of $P$ about line $L$
Solution
Coordinate of point $R$ are $(2 \lambda+1,3 \lambda-2,6 \lambda+7)$ $\overrightarrow{P R}=\langle 2 \lambda-1,3 \lambda+3,6 \lambda+7\rangle$
From diagram $\overrightarrow{P R} \cdot(2 \hat{i}+3 \hat{j}+6 \hat{k})=0$
$4 \lambda-2+9 \lambda+9+36 \lambda+42=0$
$49 \lambda+49=0 \Rightarrow \lambda=-1$
So, coordinates of $R$ are (-1,-5,1)
$\Rightarrow $ coordinates of $Q$ are (-4,-5,2)