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Question #40804

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(x,y)$ be a moving point in the first quadrant on a hyperbola $H$ with its center at the origin and transverse axis along the x-axis. The distance of $P$ to the conjugate axis is $x$ . Let $a$ be the semi-transverse axis of $H$ . It is given that $\lim_{y \to 0} \frac{x - a - \frac{3}{2}y^2}{y^4} = -\frac{27}{8}$ Find the length of the latus rectum of the hyperbola $H$ .

Options

A.
$\frac{2}{3}$
B.
$\frac{4}{3}$
C.
$\frac{1}{3}$
D.
$\frac{8}{9}$

Check Your Answer

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

Hyperbola Conjugate Axes|Locus of a Moving Point|The L'Hôpital's Rule Misuse

Why This Question Is Tricky

Common Mistakes

This question baits students into applying L'Hôpital's rule on a 0/0 indeterminate form involving radicals.

Blindly differentiating the radical expression $\sqrt{a^2 + \frac{a^2}{b^2}y^2}$ multiple times without factoring out $y$ at each step leads to an escalating algebraic nightmare and potential division by zero errors.

The mathematically elegant path forces the use of the Binomial expansion, which immediately reveals the coefficients of the locus equation and bypasses the derivative trap entirely.

Quick Facts About This Question

Question Information

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