3D Geometry
Double Image in Lines — a+b
nta_pyq_2026_jan
Grade 12
Question:
If the image of the point $P(a,2,a)$ in the line $\dfrac{x}{2}=\dfrac{y+a}{1}=\dfrac{z}{1}$ is $Q$ and the image of $Q$ in the line $\dfrac{x-2b}{2}=\dfrac{y-a}{1}=\dfrac{z+2b}{-5}$ is $P$, then $a+b$ is equal to _____.
Step-by-Step Solution
Key Concept: Find image of $P(a,2,a)$ in line 1: foot at $(2\lambda,\lambda-a,\lambda)$. Perpendicularity: $(a-2\lambda,2-\lambda+a,a-\lambda)\cdot(2,1,1)=0$. Then image $Q$. Reflect $Q$ back to $P$ in line 2.
$a+b=3$.
Correct Answer: 3