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Question #71556

Difficulty: 8/10 | 🟢 Free

The Question

Let $f(x) = \sqrt{x - 2a}$ for $x \ge 2a$ , where $a > 0$ . Let $h(x) = f^{-1}(f(x))$ . Consider the family of curves $\Gamma_1$ satisfying the differential equation $\frac{dy}{dx} = \frac{-1}{2h(x) - 2a}$ . Let $\Gamma_2$ be the orthogonal trajectory of $\Gamma_1$ that passes through the point $(3a, 0)$ . Let $R$ be the sum of the $x$ -coordinates of all points where $\Gamma_2$ intersects the $x$ -axis. Let $H$ be the number of points where $\Gamma_2$ has a horizontal tangent. Find the value of $R - aH$ .

Options

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

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Orthogonal Trajectories|Rolle's Theorem Root Existence|Composition Inverse Identity Traps|Inverse Existence Criterion

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 'Inverse Identity Trap' combined with orthogonal trajectories and the existence of roots/horizontal tangents (Rolle's Theorem conditions).

Students commonly assume $f^{-1}(f(x)) = x$ globally, ignoring that the identity only holds within the domain of the inner function $f(x)$ (here, $x \ge 2a$ ).

This false assumption leads to accepting 'ghost roots' ( $x = -a$ ) and falsely applying Rolle's Theorem to find non-existent horizontal tangents ( $x = a$ ) that lie outside the valid domain of the orthogonal trajectory.

By forcing the evaluation of $R - aH$ , the problem heavily penalizes the failure to apply global domain constraints to local derivative properties.

Quick Facts About This Question

Question Information

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