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Question #72103

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the region bounded by the locus of all curves passing through the origin and satisfying the differential equation $(a^2 - x^2) \left( \frac{dy}{dx} \right)^2 = b^2 (4x^4 - 4a^2 x^2 + a^4)$ where $a, b > 0$ . The area of the region $S$ is:

Options

A.
$\frac{4}{3} a^3 b$
B.
$\frac{2}{3} a^3 b$
C.
$\frac{1}{3} a^3 b$
D.
0

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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

Implicit to Explicit Conversion|Axis of Integration Selection|The Sign-Consistency Principle

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 convert a non-linear differential equation into an explicit family of curves and strategically select the axis of integration.

The primary trap lies in the 'Sign-Consistency Principle'.

When taking the square root of the DE, two branches $y = \pm bx\sqrt{a^2 - x^2}$ emerge.

Students often blindly subtract these branches and integrate over the entire domain $[-a, a]$ , yielding an area of 0.

They fail to recognize that the branches cross at the origin, meaning the 'upper' and 'lower' functions swap.

A rigorous sign-test is required to split the integral at $x=0$ (or use symmetry and absolute values), while attempting to integrate along the y-axis leads to an algebraically intractable inverse function.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Implicit to Explicit Conversion
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free