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Question #41596

Difficulty: 8/10 | 🟢 Free

The Question

Let a curve $C$ satisfy the differential equation $(2x - y - 1)dx + (2y - x - 1)dy = 0$ . If the curve passes through the point $(3, 2)$ , then the area of the region enclosed by $C$ is

Options

A.
$2\sqrt{3}\pi$
B.
$\frac{14\pi}{\sqrt{3}}$
C.
$\frac{4\pi}{\sqrt{3}}$
D.
$\frac{8\pi}{\sqrt{3}}$

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

Homogeneous Function Homogeneity|Standard Ellipse/Circle Area|Shifting Origin

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply boundary conditions after a coordinate transformation in a reducible-to-homogeneous differential equation.

The primary trap lies in evaluating the constant of integration.

Students often substitute the original coordinates $(3,2)$ directly into the homogeneous form $X^2 - XY + Y^2 = K$ , yielding an incorrect constant $K=7$ (Option B).

Alternatively, if solved via exact differentials, students obtain $x^2 - xy + y^2 - x - y = 2$ and frequently use the RHS value $K=2$ directly in the area formula without completing the square to account for the origin shift, leading to Option C.

The correct geometric interpretation requires evaluating the constant in the shifted coordinate system $(X,Y) = (2,1)$ to find the true ellipse parameter $K=3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Homogeneous Function Homogeneity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free