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Question #40479

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{z \in \mathbb{C} : |z - 4| + |z + 4| = 10\}$ . For any $z \in S$ , let $w(z)$ be the unique complex number on the circle $|w| = 5$ such that $\text{Re}(z) = \text{Re}(w)$ and $\text{Im}(z) = \frac{3}{5}\text{Im}(w)$ . If $z_1, z_2 \in S$ satisfy $|w(z_1) - w(z_2)| = 6$ , find the maximum value of $|z_1 - z_2| + |z_1 + z_2|$ .

Options

A.
$2\sqrt{34}$
B.
$\frac{58}{5}$
C.
$\frac{54}{5}$
D.
$4\sqrt{13}$

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

Triangle Inequality Mind Games|Ellipse Auxiliary Circle|Algebraic-Trigonometric Graphical Mapping

Why This Question Is Tricky

Common Mistakes

This question tests the algebraic-trigonometric mapping of complex numbers on an ellipse to its auxiliary circle, combined with a constrained optimization of the Triangle Inequality.

The primary trap lies in assuming the maximum of $|z_1 - z_2| + |z_1 + z_2|$ occurs at the geometric extremities (the major or minor axes, corresponding to $\alpha = 0$ or $\pi/2$ ).

These endpoints yield the ghost solutions $58/5$ and $54/5$ .

By parameterizing the moduli and applying the Cauchy-Schwarz inequality, students must verify that the true maximum occurs at an interior point of the restricted trigonometric domain $x \in [3,5]$ , rewarding rigorous algebraic optimization over naive geometric intuition.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Triangle Inequality Mind Games
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free