3D Geometry
Line-Plane Angle — Finding α+2β+6γ
nta_pyq_2023_apr
Grade 12
Question:
Line $L:\ x=\frac{1-y}{-2}=\frac{z-3}{\lambda}$ meets plane $x+2y+3z=4$ at $(\alpha,\beta,\gamma)$. Angle between $L$ and plane is $\cos^{-1}(\sqrt{\frac{5}{14}})$. Then $\alpha+2\beta+6\gamma$ is equal to
Step-by-Step Solution
Key Concept: Direction of $L$: $(1,-2,\lambda)$. Normal $(1,2,3)$. $\sin\theta=\frac{|1-4+3\lambda|}{\sqrt{1+4+\lambda^2}\cdot\sqrt{14}}$. Use given angle to find $\lambda$.
$11$.
Correct Answer: 11