<p>Given \(|z - 3 + 2i| \leq 4\), find \(|z|_{\max} - |z|_{\min}\).</p>
Step-by-Step Solution
Key Concept: The constraint |z - 3 + 2i| ≤ 4 represents a closed disk centered at (3, -2) with radius 4. The maximum and minimum values of |z| occur at points on this disk that are collinear with the origin and the center.
<p><strong>Step 1:</strong> Interpret the constraint. The equation |z - 3 + 2i| ≤ 4 represents a closed disk with center C = 3 - 2i and radius r = 4.</p><p><strong>Step 2:</strong> Find distance from origin to center: |C| = |3 - 2i| = √(9 + 4) = √13.</p><p><strong>Step 3:</strong> Since √13 ≈ 3.6 < 4, the origin lies inside the disk.</p><p><strong>Step 4:</strong> When origin is inside the disk, maximum |z| occurs at the far point of the disk from origin along the line from O through C: |z|_max = |C| + r = √13 + 4.</p><p><strong>Step 5:</strong> Minimum |z| occurs at the near point on the disk (on the opposite side of C from the far point): |z|_min = r - |C| = 4 - √13.</p><p><strong>Step 6:</strong> Calculate the difference: |z|_max - |z|_min = (√13 + 4) - (4 - √13) = 2√13.</p><p>∴ Answer: <strong>2√13</strong></p>
Correct Answer: A