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Question #41599

Difficulty: 8/10 | 🟢 Free

The Question

A moving point $P(x,y)$ in the first quadrant traces a curve $C$ passing through the point $\left(\frac{1}{2}, \frac{\sqrt{7}}{2}\right)$ . The slope of the tangent to $C$ at any point $P$ is given by the differential equation $\frac{dy}{dx} = \frac{y^2 - x^2}{2xy}$ . Furthermore, the locus of $P$ is constrained such that its distance from the origin is strictly less than its distance from the line $x = 3$ . Let $S$ be the set of all possible $y$ -coordinates of the point $P$ . If the supremum of $S$ is $M$ , what is the value of $M^2$ ?

Options

A.
2
B.
3
C.
4
D.
5

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

Substitution Logic in Homogeneous DE|Locus of a Moving Point|Point-Line Relative Position

Why This Question Is Tricky

Common Mistakes

This question tests the integration of homogeneous differential equations with coordinate geometry and locus constraints.

The primary trap is the 'Lethal Collision' boundary condition: students often solve the DE to find the family of circles ( $x^2+y^2=4x$ ) but fail to restrict the domain of the locus based on the point-line distance inequality.

Assuming the entire curve is valid leads to finding the global maximum of the circle ( $y=2$ at $x=2$ ), yielding an incorrect answer of 4.

The constraint forces students to reject branches of the solution, testing their grasp on the boundaries of the domain of existence.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Differential Equations - Class 12
Concept	Substitution Logic in Homogeneous DE
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free