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Question #40894

Difficulty: 8/10 | 🟢 Free

The Question

A variable chord $PQ$ of the hyperbola $H: x^2 - y^2 = 8$ is divided externally by a point $R$ in the ratio $2:1$ . If $R$ always lies on the asymptote $y = x$ of the hyperbola, then the line containing the chord $PQ$ is always tangent to which of the following curves?

Options

A.
$x^2 - y^2 = 9$
B.
$x^2 - y^2 = 8$
C.
$y^2 - x^2 = 1$
D.
$x^2 - y^2 = 1$

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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

Section Formula Internally/Externally|Condition of Tangency|Asymptote Geometry

Why This Question Is Tricky

Common Mistakes

This question tests the application of the external section formula on a hyperbola's chord and its relation to the condition of tangency.

The core trap lies in conflating the external section point $R$ with the intersection of the asymptotes (the center of the hyperbola).

Students often incorrectly assume that since $R$ lies on the asymptote, it must be the origin $(0,0)$ , leading them to erroneously conclude $c=0$ and apply a false tangency condition.

Furthermore, algebraic sign errors in the section formula (using internal division $2:1$ instead of external) will lead to the incorrect envelope $y^2 - x^2 = 1$ , punishing superficial memorization.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Section Formula Internally/Externally
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free