3D Geometry
Angle between planes; line on plane
nta_pyq_2023_jan
Grade 12
Question:
Let $\theta$ be the angle between the planes $P_1 = \vec{r}\cdot(\hat{i}+\hat{j}+2\hat{k})=9$ and $P_2 = \vec{r}\cdot(2\hat{i}-\hat{j}+\hat{k})=15$. Let L be the line that meets $P_2$ at the point $(4, -2, 5)$ and makes an angle $\theta$ with the normal of $P_2$. If $\alpha$ is the angle between L and $P_2$ then $(\tan^2\theta)(\cot^2\alpha)$ is equal to _____.
Step-by-Step Solution
Key Concept: Find $\theta$ using dot product of normals; if line makes angle $\theta$ with normal then $\alpha = 90°-\theta$; compute $\tan^2\theta \cdot \cot^2\alpha$.
$\theta=\pi/3$, $\alpha=\pi/6$. $(\tan^2\theta)(\cot^2\alpha) = 3\times3 = 9$. Answer: 9
Correct Answer: 9