3D Geometry
Distance from a point to a line in 3D
nta_pyq_2023_jan
Grade 12
Question:
Let a line L pass through the point $P(2, 3, 1)$ and be parallel to the line $x + 3y - 2z - 2 = 0 = x - y + 2z$. If the distance of L from the point $(5, 3, 8)$ is $\alpha$, then $3\alpha^2$ is equal to _____.
Step-by-Step Solution
Key Concept: Find direction of L as cross product of plane normals, then compute distance from given point to line L.
Direction $(1,-1,-1)$. Line through $(2,3,1)$: point $(k+2,-k+3,-k+1)$. From $Q=(5,3,8)$: $QR$ perp to direction, $k=-4/3$. $\alpha^2 = 474/9$. $3\alpha^2 = 158$. Answer: 158
Correct Answer: 158