JEE Main 2026 — Three Dimensional Geometry Question with Solution

Let $M_1$ be the foot of the perpendicular from $P(0,-5,0)$ to the first line $L_1:\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{-2}$.
A general point on $L_1$ is $M_1(2\lambda +1,\lambda ,-2\lambda -1)$.
The direction ratios of $P{M}_1$ are $(2\lambda +1,\lambda +5,-2\lambda -1)$.
Since $P{M}_1$ is perpendicular to $L_1$, their dot product is zero:
$2(2\lambda +1)+1(\lambda +5)-2(-2\lambda -1)=0$
$4\lambda +2+\lambda +5+4\lambda +2=0\Rightarrow 9\lambda +9=0\Rightarrow \lambda =-1$
Thus, $M_1$ is $(-1,-1,1)$.

Let $M_2$ be the foot of the perpendicular from $Q\left(0,\frac{-1}{2},0\right)$ to the second line $L_2:\frac{x-1}{-1}=\frac{y+9}{4}=\frac{z+1}{1}$.
A general point on $L_2$ is $M_2(-\mu +1,4\mu -9,\mu -1)$.
The direction ratios of $Q{M}_2$ are $\left(-\mu +1,4\mu -\frac{17}{2},\mu -1\right)$.
Since $Q{M}_2$ is perpendicular to $L_2$, their dot product is zero:
$-1(-\mu +1)+4\left(4\mu -\frac{17}{2}\right)+1(\mu -1)=0$
$\mu -1+16\mu -34+\mu -1=0\Rightarrow 18\mu -36=0\Rightarrow \mu =2$
Thus, $M_2$ is $(-1,-1,1)$.

Since $M_1$ and $M_2$ are the same point $M(-1,-1,1)$, the diagonals $PR$ and $QS$ of the quadrilateral $PQRS$ bisect each other at $M$. Therefore, $PQRS$ is a parallelogram.

The area of the parallelogram is given by $\frac{1}{2}|\overrightarrow{PR}\times \overrightarrow{QS}|$.
We have:
$\overrightarrow{PR}=2\overrightarrow{PM}=2(-1-0,-1-(-5),1-0)=(-2,8,2)$
$\overrightarrow{QS}=2\overrightarrow{QM}=2\left(-1-0,-1-\left(\frac{-1}{2}\right),1-0\right)=(-2,-1,2)$

Now, compute the cross product $\overrightarrow{PR}\times \overrightarrow{QS}$:
$\overrightarrow{PR}\times \overrightarrow{QS}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ -2& 8& 2\\ -2& -1& 2\end{array}\right|=\hat{i}(16-(-2))-\hat{j}(-4-(-4))+\hat{k}(2-(-16))=18\hat{i}+18\hat{k}$

The area of the parallelogram is:
$\text{Area}=\frac{1}{2}|18\hat{i}+18\hat{k}|=\frac{1}{2}\sqrt{18^2+18^2}=\frac{18\sqrt{2}}{2}=9\sqrt{2}$

The square of the area is $(9\sqrt{2}{)}^2=162$.

Answer: $162$