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Question #72142

Difficulty: 8/10 | 🟢 Free

The Question

Let a ray $L$ in the region $x > 0$ be defined as the intersection of the plane $x=y$ and the locus of a point $P(x,y,z)$ whose distance from the plane $x+y+z=0$ is equal to $\alpha(x+y-z)$ , where $\alpha > \frac{1}{\sqrt{3}}$ . If $\theta$ is the acute angle between the ray $L$ and the $xy$ -plane, then $\sin \theta$ is equal to:

Options

A.
$\frac{\sqrt{2}(\alpha\sqrt{3} - 1)}{\sqrt{9\alpha^2 - 2\alpha\sqrt{3} + 3}}$
B.
$\frac{\sqrt{2}(\alpha\sqrt{3} + 1)}{\sqrt{9\alpha^2 + 2\alpha\sqrt{3} + 3}}$
C.
$\frac{\alpha\sqrt{3} - 1}{\sqrt{9\alpha^2 - 2\alpha\sqrt{3} + 3}}$
D.
$\frac{\alpha\sqrt{3} + 1}{\sqrt{9\alpha^2 + 2\alpha\sqrt{3} + 3}}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Angle between line and plane|Distance from point to plane|Modulus Sign Dependency

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the student's ability to handle modulus sign dependencies in 3D geometry.

When equating a distance (which inherently involves an absolute value) to a variable expression, an implicit domain constraint arises (here, $x+y-z \ge 0$ ).

Squaring both sides to solve the equation introduces an extraneous root that corresponds to the negative branch of the modulus.

Students who blindly square the equation without checking the domain constraint $m \le 2$ will incorrectly select the slope $m_1$ , leading them straight to the trap option B.

Recognizing that the modulus dictates the valid geometric region is crucial for identifying the correct direction vector of the ray.

Quick Facts About This Question

Question Information

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