3D Geometry
Image in Line — a²+b²+c²
nta_pyq_2026_jan
Grade 12
Question:
If the image of the point $P(1,2,a)$ in the line $\dfrac{x-6}{3}=\dfrac{y-7}{2}=\dfrac{7-z}{2}$ is $Q(5,b,c)$, then $a^2+b^2+c^2$ is equal to
Step-by-Step Solution
Key Concept: Line direction $(3,2,-2)$, passes through $(6,7,7)$. Foot $M$: $(6+3t,7+2t,7-2t)$. $\ vec{PM}\cdot(3,2,-2)=0$. From $Q_x=5$: $11+6t=5\Rightarrow t=-1$.
$a^2+b^2+c^2=298$.
Correct Answer: 1