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Question #41343

Difficulty: 8/10 | 🟢 Free

The Question

Let $H$ be the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$ . A point $P(x, y)$ moves on the right branch of $H$ such that $y > 0$ . Let $L$ be the polar of $P$ with respect to the conjugate hyperbola of $H$ . If the slope of $L$ is denoted by $m$ , then $y$ can be expressed as a function of $m$ , say $y = g(m)$ . Find the value of $-64 \cdot g'\left(\frac{20}{9}\right)$ .

Options

A.
135
B.
-135
C.
144
D.
243

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

Inverse Function Derivative|Polar and Pole Duality|Hyperbola Conjugate Axes|Reflective Property

Why This Question Is Tricky

Common Mistakes

This problem tests the synthesis of polar/pole duality with inverse function differentiation.

The core trap lies in the branch restriction of the hyperbola.

Students often implicitly differentiate the hyperbola equation $16x^2 - 9y^2 = 144$ to find $dx/dy$ , but fail to correctly map the slope of the polar line $m = 16x/9y$ to a single-variable function restricted to the right branch ( $x > 0$ ).

If they ignore the branch constraint, they might take the negative root for $x$ , inverting the sign of the derivative.

Furthermore, the duality between the hyperbola and its conjugate means the polar's slope matches the tangent's slope, tempting students to incorrectly compute $dx/dm$ instead of $dy/dm$ , leading to the distractor 243.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Inverse Function Derivative
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free