3D Geometry
3D Coordinates — Distance from Axis Using Angles
nta_pyq_2024_apr
Grade 12
Question:
Let $P(x,y,z)$ be a point in the first octant, whose projection in the $xy$-plane is the point $Q$. Let $OP=\gamma$; the angle between $OQ$ and the positive $x$-axis be $\theta$; and the angle between $OP$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is
$\gamma\sqrt{1-\sin^2\phi\cos^2\theta}$
$\gamma\sqrt{1-\sin^2\theta\cos^2\phi}$
$\gamma\sqrt{1+\cos^2\phi\sin^2\theta}$
$\gamma\sqrt{1+\cos^2\theta\sin^2\phi}$
Step-by-Step Solution
Key Concept: $x=\gamma\sin\phi\cos\theta$, $y=\gamma\sin\phi\sin\theta$, $z=\gamma\cos\phi$. Distance from $x$-axis $=\sqrt{y^2+z^2}=\sqrt{\gamma^2\sin^2\phi\sin^2\theta+\gamma^2\cos^2\phi}$.
Distance from $x$-axis $=\gamma\sqrt{1-\sin^2\phi\cos^2\theta}$.
Correct Answer: 1