3D Geometry
Perpendicular Line — Square of Parallelogram Area
nta_pyq_2026_jan
Grade 12
Question:
Let a line $L$ passing through the point $P(1,1,1)$ be perpendicular to the lines $\dfrac{x-4}{4}=\dfrac{y-1}{1}=\dfrac{z-1}{1}$ and $\dfrac{z-17}{1}=\dfrac{y-71}{1}=\dfrac{z}{0}$. Let the line $L$ intersect the $yz$-plane at the point $Q$. Another line parallel to $L$ and passing through the point $S(1,0,-1)$ intersects the $yz$-plane at the point $R$. Then the square of the area of the parallelogram $PQRS$ is equal to _____.
Step-by-Step Solution
Key Concept: Direction of $L=(4,1,1)\times(1,1,0)=(-1,1,3)$. Line $L$: $(x,y,z)=(1,1,1)+t(-1,1,3)$. Intersection with $yz$-plane ($x=0$): $t=1$, $Q=(0,2,4)$.
Square of area $=6$.
Correct Answer: 6