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Question #41610

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_1$ be a curve defined by the differential equation $(x^2+4)dy + (2xy - 24)dx = 0$ , passing through the point $(0, 12)$ . Let $P$ be the point of inflection of $C_1$ having the largest $x$ -coordinate. A hyperbola $H$ , with its transverse axis along the $x$ -axis, has an asymptote in the first quadrant whose slope is equal to the absolute value of the slope of the tangent to $C_1$ at $P$ . What is the eccentricity of the hyperbola $H$ ?

Options

A.
$\sqrt{5}$
B.
$\sqrt{10}$
C.
$\sqrt{17}$
D.
$\sqrt{145}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Exact Differential Recognition|Asymptote Geometry|Point of Inflection Conditions

Why This Question Is Tricky

Common Mistakes

This question bridges exact differential equations with conic sections, specifically testing the procedural fluency of finding inflection points and the conceptual understanding of asymptotes.

$The primary trap lies in the misinterpretation of the inflection point's properties: students often erroneously use the x-coordinate (yielding e=\sqrt{5}) or y-coordinate (yielding e=\sqrt{145}) of the inflection point as the slope, or confuse the horizontal asymptote of the curve C_1 (y=0) with the asymptote of the hyperbola.$

By requiring the absolute value of the derivative at the specific root of y''=0 that has the largest x-coordinate, the problem ensures that only a complete, rigorous application of calculus and coordinate geometry yields the correct eccentricity.

Quick Facts About This Question

Question Information

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