3D Geometry
Feet of Perpendiculars — Right Angle Condition
nta_pyq_2024_jan
Grade 12
Question:
Let $Q$ and $R$ be the feet of perpendiculars from the point $P(a,a,a)$ on the lines $x=y,z=1$ and $x=-y,z=-1$ respectively. If $\angle QPR$ is a right angle, then $12a^2$ is equal to
Step-by-Step Solution
Key Concept: Line 1: $x=y,z=1$ — parametrize as $(t,t,1)$. Foot $Q$: direction $(1,1,0)\perp(Q-P)$. Line 2: $x=-y,z=-1$ — parametrize as $(s,-s,-1)$. Foot $R$: direction $(1,-1,0)\perp(R-P)$.
$a^2=1$. $12a^2=12$.
Correct Answer: 12