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Question #68487

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be the curve satisfying the differential equation $(x^3 - 3xy^2) dx + (3x^2 y - y^3) dy = 0$ and passing through the point $(a, 0)$ , where $a > 0$ . The area of the region enclosed by the loop of $C$ in the right half-plane ( $x \ge 0$ ) is:

Options

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

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

Bounded vs Unbounded Regions|Homogeneous Function Homogeneity|Even and Odd Function Symmetry

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 resolution of a homogeneous differential equation that reduces to a classic polar curve (the Lemniscate of Bernoulli).

The primary trap lies in the partial fraction decomposition and the subsequent symmetry assumption.

A common algebraic error during integration (e.g., mismanaging the numerator $3v-v^3$ ) leads students to the equation of a circle $x^2+y^2=ax$ , resulting in the incorrect area $\pi a^2/4$ .

Furthermore, students who correctly identify the Lemniscate often fail to establish the correct angular bounds for the right half-plane loop ( $\theta \in [-\pi/4, \pi/4]$ ), mistakenly calculating only the first quadrant area ( $a^2/4$ ) or integrating over an invalid domain.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Bounded vs Unbounded Regions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free