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Question #40840

Difficulty: 8/10 | 🟢 Free

The Question

Consider a hyperbola $H$ in the Argand plane. One of the asymptotes of $H$ is the line $L$ given by $3(z+\bar{z}) + 4i(z-\bar{z}) = 0$ . A region $S$ is defined as the set of all points $z$ such that the distance from $z$ to the focus $z_0 = 1+2i$ of $H$ is less than or equal to its distance to the asymptote $L$ . Find the minimum value of $|z|^2$ for $z \in S$ .

Options

A.
1
B.
2
C.
4
D.
5

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

Asymptote Geometry|Geometrical Interpretation of Modulus|Locus Problems via Transformation

Why This Question Is Tricky

Common Mistakes

This question tests the geometrical interpretation of complex loci and optimization.

The primary trap is the assumption that a locus involving a hyperbola's focus and asymptote defines the hyperbola itself.

Instead, setting the eccentricity factor to 1 with respect to the asymptote defines a parabolic region.

Students must recognize this degenerate mapping and use coordinate rotation to minimize the distance to the origin, bridging complex numbers with advanced coordinate geometry.

Quick Facts About This Question

Question Information

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