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Question #74475

Difficulty: 8/10 | 🟢 Free

The Question

Let the equation of a plane $\Pi$ be $lx + my + nz = p$ , where $p$ is the perpendicular distance of the plane from the origin ( $p > 0$ ). It is given that $l = \frac{2}{\pi} \sin^{-1} \alpha$ , $m = \frac{2}{\pi} \cos^{-1} \alpha$ , and $n = \frac{1}{\sqrt{2}}$ for some real $\alpha$ . The plane $\Pi$ passes through the point $P(2, 2, -\sqrt{2})$ .

A line $L: \frac{x - 4}{a} = \frac{y - \beta}{b} = \frac{z - \sqrt{2}}{-2}$ lies entirely in the plane $\Pi$ . If the line $L$ is perpendicular to the vector $\hat{i} - \hat{j}$ , then the value of $a^2 + b^2 + \beta^2 + 2\alpha^2$ is:

Options

A.
21
B.
25
C.
19
D.
23

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

Coplanarity Condition|Line lying in a plane|Normal Form Parameters|Domain Constraint Intersections

Why This Question Is Tricky

Examiner Levers

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

The question tests the student's understanding of the normal form of a plane equation and the coplanarity conditions for a line.

The primary trap is the assumption that $l, m, n$ in $lx+my+nz=p$ are mere direction ratios.

By explicitly defining $p$ as the perpendicular distance from the origin, the coefficients are forced to be direction cosines, invoking the hidden constraint $l^2+m^2+n^2=1$ .

This constraint intersects with the domain and range properties of inverse trigonometric functions to uniquely determine the plane's normal vector.

Students who miss this will be unable to find $\alpha$ and will fail the subsequent coplanarity tests.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Coplanarity Condition
Difficulty	8
Cognitive Load	7
Target Solve Time	170 sec
Access	Free