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Question #40819

Difficulty: 8/10 | 🟢 Free

The Question

Let $H$ be a hyperbola whose horizontal and vertical asymptotes are given by the lines $y = 22$ and $x = 22$ , respectively. Let $D$ be the diameter of $H$ that bisects a family of parallel chords having slope $m$ . The value of $m$ is equal to $\left. \frac{dy}{dx} \right|_{(1,1)}$ for the implicit curve $y^2 - 3y + 2x^{136} = 0$ . If the diameter $D$ intersects the $y$ -axis at the point $(0, \beta)$ , find the value of $\beta$ .

Options

A.
6006
B.
-5962
C.
5984
D.
-6006

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

Diameter of Conics|Asymptote Geometry|Rational Function Asymptotes|Implicit Differentiation Rate

Why This Question Is Tricky

Common Mistakes

This question tests the conceptual bridge between implicit differentiation and the geometry of conics, specifically the locus definition of a diameter.

The primary trap lies in the slope of the diameter for a rectangular hyperbola.

Students often incorrectly assume the diameter shares the slope of the parallel chords ( $m$ ) or make a sign error when deriving the locus of midpoints, leading to the incorrect equation $y - 22 = 272(x - 22)$ and the trap answer -5962.

By forcing the calculation of $m=272$ via implicit differentiation, the problem ensures students cannot rely on standard form memorization and must reconcile the derived rate with the asymptotic center (22,22).

Quick Facts About This Question

Question Information

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