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Question #40820

Difficulty: 8/10 | 🟢 Free

The Question

Let $E$ be an ellipse in the complex plane given by the equation $|z - 2 - i| + |z - 2 + i| = 4$ . If $z_1$ and $z_2$ are the endpoints of any diameter of the ellipse $E$ , then the range of the expression $|z_1|^2 + |z_2|^2$ is

Options

A.
$[14, 16]$
B.
$[6, 8]$
C.
$[12, 16]$
D.
$[20, 24]$

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

Diameter of Conics|Lagrange Identity|Geometry of Complex Inequalities|Position Vector Origin Dependence

Why This Question Is Tricky

Common Mistakes

This question tests the application of the parallelogram law (Lagrange's identity in the complex plane) to relate the coordinates of a conic's diameter to its geometric length.

The core trap lies in 'Position Vector Origin Dependence'.

Students often shift the origin to the ellipse's center to simplify the equation, correctly finding the range of $|Z_1|^2 + |Z_2|^2$ as $[6, 8]$ in the new frame.

However, they fail to realize that the sum of squared moduli is not translation-invariant, forgetting to apply the inverse transformation $z = Z + 2$ , which leads them to the incorrect option $[6, 8]$ .

Quick Facts About This Question

Question Information

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