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Question #40832

Difficulty: 8/10 | 🟢 Free

The Question

Let the rectangular hyperbola $H$ be given by the equation $x^2 - y^2 - 4x + 6y - 9 = 0$ . Let $f: (-\infty, 0] \to [3, \infty)$ be a bijective function whose graph is a branch of $H$ . If $g(x)$ is the inverse function of $f(x)$ , find the value of $g'(3+\sqrt{5})$ .

Options

A.
$\frac{\sqrt{5}}{3}$
B.
$-\frac{\sqrt{5}}{3}$
C.
$\frac{3}{\sqrt{5}}$
D.
$-\frac{3}{\sqrt{5}}$

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

Rectangular Hyperbola|Shifting Origin|Bijectivity and Inverse|Domain and Range Nuance

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle the inverse of a non-bijective function by strictly adhering to a restricted domain.

The primary trap lies in solving $(x-2)^2 = 9$ and instinctively choosing the positive root $x=5$ , which belongs to the other branch of the hyperbola.

Using $x=5$ yields the incorrect derivative value of positive $\sqrt{5}/3$ .

The problem forces candidates to distinguish between the global algebraic curve and the specific bijective branch defined by the domain $(-\infty, 0]$ , ensuring they evaluate the derivative at the correct coordinate.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Rectangular Hyperbola
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free