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Question #41688

Difficulty: 8/10 | 🟢 Free

The Question

Let $z = x + iy$ be a complex number satisfying $x \ge 0$ , $y \ge 0$ , and the inequality $|z - 2 - 4i| + |iz + 12 + 4i| \le 10$ . Find the minimum value of the objective function $f(z) = |z - 2 - 4i|^2 + |iz + 12 + 4i|^2$ .

Options

A.
50
B.
$\frac{500}{9}$
C.
100
D.
$\frac{400}{9}$

Check Your Answer

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

Non-Negativity Restriction|Triangle Inequality Mind Games|Algebraic Manipulation of i

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of algebraic manipulation of complex loci and linear programming constraints.

The primary trap is twofold: First, students often incorrectly simplify $|iz + 12 + 4i|$ by dropping the $i$ or misattributing the coordinates, leading them to the wrong locus (e.g., a segment to $12+4i$ ).

Second, even if they find the correct segment, the objective function $f(t)$ is a quadratic whose unconstrained minimum occurs at $t=1/2$ (yielding the trap answer 50).

Students accustomed to standard LP problems might ignore the non-negativity restriction $x \ge 0$ , failing to realize that the unconstrained minimum lies outside the first quadrant, forcing the true minimum to occur at the boundary $t=1/3$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Linear Programming - Class 12
Concept	Non-Negativity Restriction
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free