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Question #41684

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the region in the complex plane consisting of all $z = x+iy$ satisfying the following system of inequalities: $|z - 2| \le |z + 2|$ $|z - 2i| \le |z + 2i|$ $|z| \ge |z - 2 - 2i|$ $|z - 4| \le |z - 8|$ $|z - 4i| \le |z - 8i|$ $|z - 3 - 3i| \le |z - 5 - 5i|$ Let $f(z) = \text{Re}((1-i)z)$ . If $M$ is the maximum value of $f(z)$ for $z \in S$ , and $E$ is the set of all points $z \in S$ such that $f(z) = M$ , find the length of the line segment representing $E$ .

Options

A.
$2\sqrt{2}$
B.
$4\sqrt{2}$
C.
$6\sqrt{2}$
D.
$8\sqrt{2}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Multiple Optimal Solutions|Condition of Collinearity|Geometry of Complex Inequalities

Why This Question Is Tricky

Common Mistakes

The question tests the geometric interpretation of complex inequalities and the condition for multiple optimal solutions in linear programming.

The primary trap is algebraic brute force: students often expand $|z-z_1| \le |z-z_2|$ algebraically, consuming time and risking errors, instead of recognizing it as the half-plane defined by the perpendicular bisector of the segment joining $z_1$ and $z_2$ .

Furthermore, the objective function $f(z) = \text{Re}((1-i)z) = x+y$ is deliberately chosen to be parallel to the boundary constraint $x+y \le 8$ .

This creates a scenario of multiple optimal solutions (a line segment rather than a unique vertex), testing the student's understanding of collinearity between the objective function's level curves and the feasible region's boundaries, a core concept in optimization.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Multiple Optimal Solutions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free