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Question #70567

Difficulty: 8/10 | 🟢 Free

The Question

Consider the family of curves $\mathcal{C}$ defined by the equation $x^2 - y^2 + 2y|x| = c$ , where $c \in \mathbb{R}$ . If the family of orthogonal trajectories to $\mathcal{C}$ is denoted by $\mathcal{T}$ , then the equation representing $\mathcal{T}$ is (where $K$ is an arbitrary real constant):

Options

A.
y^2 - x^2 + 2y|x| = K
B.
y^2 - x^2 + 2xy = K
C.
x^2 - y^2 - 2y|x| = K
D.
x^2 + y^2 + 2y|x| = K

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

Geometric Interpretation|Orthogonal Trajectories|Rectangular Hyperbola|The Modulus Trap

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

The question tests the student's ability to find orthogonal trajectories for a family of curves involving a non-differentiable modulus function.

The trap lies in blindly differentiating the equation without splitting it into branches for $x > 0$ and $x < 0$ .

If a student ignores the modulus, they will obtain a single differential equation and incorrectly conclude that the orthogonal trajectory is $y^2 - x^2 + 2xy = K$ for all $x$ .

By properly separating the branches, the student discovers that the modulus-induced sign change alters the differential equation, leading to two distinct rectangular hyperbola orientations that must be recombined using the modulus function.

Quick Facts About This Question

Question Information

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