3D Geometry
Line and Plane Intersection
Grade Class 12
Question:
A line through $(2,-2,5)$ with direction $(1,-3,2)$ meets the plane $2x-3y+4z=163$ at $P$ and the $YZ$-plane at $Q$. If $PQ = a\sqrt{b}$ where $a>3$, find $a+b$.
Step-by-Step Solution
Key Concept: Parametrize line; find $\lambda$ for plane intersection (P) and for $x=0$ (Q); compute $|PQ|$.
Line: $(2+\lambda, -2-3\lambda, 5+2\lambda)$. At $YZ$-plane: $2+\lambda=0\Rightarrow\lambda=-2$, $Q=(0,4,1)$. At $2x-3y+4z=163$: $2(2+\lambda)-3(-2-3\lambda)+4(5+2\lambda)=163$; $4+2\lambda+6+9\lambda+20+8\lambda=163$; $19\lambda+30=163$; $\lambda=7$; $P=(9,-23,19)$. $PQ=\sqrt{81+729+324}=\sqrt{1134}=9\sqrt{14}$. $a=9>3$, $b=14$. $a+b=23$.
Correct Answer: 23