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Question #40510

Difficulty: 8/10 | 🟢 Free

The Question

Let $S = \{ z \in \mathbb{C} : |z - 3| - |z + 3| < 6 \text{ and } |z| \le 5 \}$ . Find the number of elements in $S$ such that both the real and imaginary parts of $z$ are integers.

Options

A.
75
B.
78
C.
81
D.
84

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

Geometry of Complex Inequalities|Hyperbola Conjugate Axes|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

The problem tests the student's understanding of the triangle inequality applied to complex numbers, specifically the boundary cases of $|z-z_1| - |z-z_2| \le |z_1-z_2|$ .

A common trap is assuming the strict inequality $|z-3| - |z+3| < 6$ defines a standard 2D region bounded by a hyperbola, leading students to either guess the area or incorrectly evaluate the integer points.

By recognizing that the maximum possible difference is exactly the distance between the foci (6), students must deduce that the inequality holds everywhere except on the degenerate hyperbola branch (the ray $x \le -3$ ).

Failing to account for the modulus sign dependency and the exact piecewise nature of this degenerate locus typically leads to an unadjusted count of 81, or an incorrect subtraction of both rays yielding 75.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Complex Numbers and Quadratic Equations - Class 11
Concept	Geometry of Complex Inequalities
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free