3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
Let PM be perpendicular from the point $P(1, 2, 3)$ to $x$-$y$ plane. If $\overrightarrow{OP}$ makes an angle $\theta$ with the positive direction of $z$-axis and $\overrightarrow{OM}$ makes an angle $\phi$ with the positive direction of $x$-axis, where $O$ is the origin and $\theta$ and $\phi$ are acute angles, then:
\tan \theta = \frac{\sqrt{5}}{3}
\sin \theta \sin \phi = \frac{2}{\sqrt{14}}
\tan \phi = 2
\cos \theta \cos \phi = \frac{1}{\sqrt{14}}
Step-by-Step Solution
Key Concept: Convert Cartesian coordinates to spherical coordinates using the standard transformation formulas.
Using spherical coordinates with $x = r\sin\theta\cos\phi$, $y = r\sin\theta\sin\phi$, $z = r\cos\theta$, and given $1 = r\sin\theta\cos\phi$, $2 = r\sin\theta\sin\phi$, $3 = r\cos\theta$, we find $r^2 = 1 + 4 + 9 = 14$, so $r = \sqrt{14}$. This gives $\sin\theta\cos\phi = \frac{1}{\sqrt{14}}$, $\sin\theta\sin\phi = \frac{2}{\sqrt{14}}$, $\cos\theta = \frac{3}{\sqrt{14}}$, and $\tan\phi = 2$. Also, $\tan\theta = \frac{\sqrt{5}}{3}$.
Correct Answer: Let me verify each option using the given solution:
Given: P(1, 2, 3), r = √14
From the spherical coordinate analysis:
- sin θ cos φ = 1/√14
- sin θ sin φ = 2/√14
- cos θ = 3/√14
**Option 1: