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Question #40769

Difficulty: 8/10 | 🟢 Free

The Question

Let $P$ be the point of intersection of the curves $4x^2 - y^2 = 0$ and $xy = 2$ in the first quadrant. A variable line $L$ passing through $P$ intersects the asymptotes of the hyperbola $x^2 - y^2 = 1$ at points $A$ and $B$ . If the area of $\triangle OAB$ (where $O$ is the origin) is $5$ , then the sum of the slopes of all such possible lines $L$ is:

Options

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

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Family of Lines|Asymptote Geometry|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to handle modulus dependencies within coordinate geometry, specifically when finding a family of lines bounded by asymptotic conditions.

The trap lies in the area formula of the triangle formed by a line and the asymptotes of a hyperbola, which inherently contains a modulus in the denominator: $\Delta = \frac{a b w^2}{|a^2 u^2 - b^2 v^2|}$ .

Students frequently drop the absolute value, implicitly assuming $m^2 - 1 > 0$ , which only yields the lines with slopes $|m| > 1$ .

By doing so, they completely miss the 'negative branch' ( $|m| < 1$ ), losing half of the valid family members that form the required area.

This branch discontinuity trap effectively separates rote calculators from conceptually rigorous students.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Family of Lines
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free