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Question #40831

Difficulty: 8/10 | 🟢 Free

The Question

Let $z_1, z_2 \in \mathbb{C}$ be distinct points lying on the rectangular hyperbola $z^2 + \bar{z}^2 = 4$ . If $z_1 + z_2 = 6 + 2i$ , find the value of $|z_1 - z_2|^2$ .

Options

A.
24
B.
30
C.
34
D.
40

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

Rectangular Hyperbola|Vieta’s Formulas in Complex Domain|Triangle Inequality Mind Games

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to navigate complex numbers constrained by non-convex geometry.

Students often fall into the trap of treating complex roots as free variables and misapplying real-number algebraic identities (like $|z_1 - z_2|^2 = |z_1 + z_2|^2 - 4|z_1 z_2|$ ) or the Triangle Inequality.

Another fatal error is approximating the rectangular hyperbola by its asymptotes ( $x^2 - y^2 = 0$ ), which perfectly yields the trap answer 40.

This question demands strict adherence to the hyperbola branches, anchoring the complex numbers to $x^2 - y^2 = 2$ and forcing a purely coordinate-based or rigorous Vieta's resolution.

Quick Facts About This Question

Question Information

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