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Question #41589

Difficulty: 8/10 | 🟢 Free

The Question

Let $\Gamma$ be the family of curves satisfying the differential equation $2xy \frac{dy}{dx} = y^2 - x^2 + c^2$ , where $c > 0$ is a fixed parameter. A new family of curves $\Gamma^*$ is obtained by reflecting each curve in $\Gamma$ across the line $y = c$ . Let $C \in \Gamma^*$ be a curve that is tangent to the $x$ -axis. Find the area of the region enclosed by $C$ .

Options

A.
$4\pi c^2$
B.
$\frac{4\pi c^2}{5}$
C.
$5\pi c^2$
D.
$\pi c^2$

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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

Arbitrary Constants vs Parameters|Geometric Interpretation|Reflection Principle

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between an arbitrary constant of integration and a fixed geometric parameter under a transformation.

The trap lies in the 'Reflection Principle': students often mechanically transform the differential equation or confuse the roles of the arbitrary constant $k$ and the parameter $c$ .

If they mistakenly treat $c$ as the arbitrary constant and reflect the center $(c, 0)$ across $y=k$ , they will incorrectly find the tangency condition to be $2k = \sqrt{c^2 - k^2}$ , yielding a radius of $2c/\sqrt{5}$ and leading to the distractor $4\pi c^2/5$ .

The correct approach recognizes that the reflection shifts the geometric locus of the arbitrary constant (the center of the circles) while preserving the fixed parameter's influence on the invariant radius.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Arbitrary Constants vs Parameters
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free