3D Geometry
Mirror Image of Point in a Line
nta_pyq_2024_apr
Grade 12
Question:
Consider a line $L$ passing through the points $P(1,2,1)$ and $Q(2,1,-1)$. If the mirror image of the point $A(2,2,2)$ in the line $L$ is $(\alpha,\beta,\gamma)$, then $\alpha+\beta+6\gamma$ is equal to _____
Step-by-Step Solution
Key Concept: DRs of $L$: $(-1,1,2)$. Let $C$ be foot of perpendicular from $A$ to $L$, $C=(1-t,2+t,1+2t)$. $\overrightarrow{AC}\perp L$: $(-1-t)(-1)+(t)(1)+(2t-1)(2)=0\Rightarrow6t=1\Rightarrow t=1/6$. $C=(5/6,13/6,4/3)$.
$\alpha=-1/3,\beta=7/3,\gamma=2/3$. $\alpha+\beta+6\gamma=6$.
Correct Answer: 6