3D Geometry
Mirror Image in a Line — Linear Combination
nta_pyq_2024_jan
Grade 12
Question:
Let $(\alpha,\beta,\gamma)$ be the mirror image of the point $(2,3,5)$ in the line $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$. Then $2\alpha+3\beta+4\gamma$ is equal to
Step-by-Step Solution
Key Concept: Foot of perpendicular from $(2,3,5)$ to the line: $(1+2t,2+3t,3+4t)$. Perpendicular condition: $(2t-1)2+(3t-1)3+(4t-2)4=0$. Solve for $t$, get foot $F$, mirror $=2F-P$.
$2\alpha+3\beta+4\gamma=33$.
Correct Answer: 2