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Question #41753

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the surface given by $z = \frac{17}{2}x^2 + \frac{1}{2}y^2$ , and let $P_1$ be the tangent plane to $S$ at the point $(1, 1, 9)$ . A plane $P_2$ containing the $z$ -axis intersects $P_1$ such that the acute angle $\theta$ between them satisfies $\cos \theta = \sqrt{\frac{34}{291}}$ . If $\vec{n}_2 = \cos \alpha \hat{i} + \sin \alpha \hat{j}$ is a normal vector to $P_2$ for some $\alpha \in [0, 2\pi)$ , find the sum of all possible values of $\alpha$ .

Options

A.
$4\pi + 4\tan^{-1}\left(\frac{1}{17}\right)$
B.
$2\pi + 2\tan^{-1}\left(\frac{1}{17}\right)$
C.
$4\pi - 4\tan^{-1}\left(\frac{1}{17}\right)$
D.
$2\pi - 2\tan^{-1}\left(\frac{1}{17}\right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Angle between two planes|Tangent/Normal Slope Interplay|The Sign-Consistency Principle|General Solution Principal Values

Why This Question Is Tricky

Common Mistakes

This question tests the 'Sign-Consistency Principle' in 3D geometry, specifically the bidirectional nature of normal vectors when calculating the angle between planes.

Students frequently drop the absolute value in the formula $\cos \theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1||\vec{n}_2|}$ , incorrectly assuming a unique normal orientation.

If they only solve $17 \cos \alpha + \sin \alpha = \sqrt{34}$ , they miss the obtuse normal alignments (the $-\sqrt{34}$ case) and obtain exactly half the correct sum, leading directly to the trap option B.

Additionally, the problem requires bridging Class 12 Calculus (finding the normal to a surface via partial derivatives) with vector geometry, demanding cross-domain synthesis typical of JEE Advanced.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Angle between two planes
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free