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

Answer: 3825

Given, $P\left(1,2,3\right)$
Any general point on the line is $Q\left(2\lambda +1,5\lambda +3,3\lambda +1\right)$
$\overrightarrow{\mathrm{PQ}}=2\lambda \hat{i}+\left(5\lambda +1\right)\hat{j}+\left(3\lambda -2\right)\hat{k}$
Since, $\overrightarrow{\mathrm{PQ}}$ 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 $PQ=\sqrt{400+2401+1024}=\sqrt{3825}$