Question

Let $P\left(1,2 , 3\right)$ be a point in space and $Q$ be a point on the line $\frac{x - 1}{2}=\frac{y - 3}{5}=\frac{z - 1}{3}$ such that $PQ$ is parallel to $5x-4y+3z=1$ . If the length of $PQ$ is equal to $k$ units, then the value of $k^{2}$ is equal to

Answer 1

Given, $P\left(1,2 , 3\right)$
Any general point on the line is $Q\left(2 \lambda + 1,5 \lambda + 3,3 \lambda + 1\right)$
$\overset{ \rightarrow }{P Q}=2\lambda \hat{i}+\left(5 \lambda + 1\right)\hat{j}+\left(3 \lambda - 2\right)\hat{k}$
Since, $\overset{ \rightarrow }{P Q}$ is parallel to the given plane, therefore
$10\lambda -20\lambda -4+9\lambda -6=0$
$\Rightarrow -\lambda -10=0\Rightarrow \lambda =-10$
Hence, the length of $PQ=\sqrt{400 + 2401 + 1024}=\sqrt{3825}$