3D Geometry
Angle Between Plane and Line — Distance Constraint
nta_pyq_2023_apr
Grade 12
Question:
$a,b\in\mathbb{Z}$, $|a-b|\leq10$. Angle between plane $ax+y-z=b$ and line $x-1=a-y=z+1$ is $\cos^{-1}(\frac{1}{3})$. Distance of $(6,-6,4)$ from plane is $3\sqrt{6}$. Then $a^4+b^2$ is equal to
Step-by-Step Solution
Key Concept: Line direction $(1,-1,1)$. $\sin\theta=\frac{|\vec{n}\cdot\vec{d}|}{|\vec{n}||\vec{d}|}=\frac{\sqrt{1-1/9}}{...}$. Distance condition gives $b$.
$a^4+b^2=32$.
Correct Answer: 1