JEE 2025 — Mathematics PYQ

$QR=5$ and the point $P$ be $(0,2,3)$.

Any point on line $L$ is: $5\lambda -3,2\lambda +1,3\lambda -4$
Drs of $PM\Rightarrow 5\lambda -3,2\lambda -1,3\lambda -7$
Since $L\perp PM$:
$$5(5\lambda -3)+2(2\lambda -1)+3(3\lambda -7)=0$$
$$38\lambda -38=0\Rightarrow \lambda =1$$
$$M=(2,3,-1)$$
$$PM=\sqrt{4+1+16}=\sqrt{21}$$
$$\text{Area}=\frac{1}{2}\times 5\times \sqrt{21}=\frac{m}{n}$$
$$2m-5\sqrt{21}n=0$$