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Question #41577

Difficulty: 8/10 | 🟢 Free

The Question

Let $C$ be the locus of points $(x,y)$ in the $xy$ -plane satisfying the differential equation $(4y^2 - 9x)dx - 4xy \, dy = 0$ If $C$ forms a closed loop passing through the point $(4, 3)$ , then the area of the region enclosed by $C$ is

Options

A.
$6\pi$
B.
$12\pi$
C.
$18\pi$
D.
$24\pi$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Standard Ellipse/Circle Area|Geometric Interpretation (Differential Equations)|Bernoulli Equation Transformation

Why This Question Is Tricky

Common Mistakes

This question tests the ability to identify a hidden exact differential (or apply the Bernoulli transformation), followed by the geometric interpretation of the resulting conic section.

The primary trap lies in the area calculation: students solving the DE often extract $y = \sqrt{\frac{9}{2}x - \frac{9}{16}x^2}$ due to the positive initial condition $(4,3)$ .

Integrating this function only yields the area of the upper half of the ellipse ( $6\pi$ ).

Recognizing that the locus of the differential equation forms a closed loop ( $y^2 = \dots$ ) is crucial to finding the full area ( $12\pi$ ).

Additionally, failing to properly complete the square to find the shifted standard form leads to incorrect semi-axis lengths.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Integrals - Class 12
Concept	Standard Ellipse/Circle Area
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free