<p>If <span>\(|z-3+2i| \leq 4\)</span>, then the difference between the greatest value and the least value of <span>\(|z|\)</span> is</p>
Step-by-Step Solution
Key Concept: The condition |z - (3 - 2i)| ≤ 4 describes a closed disk centered at 3 - 2i with radius 4. The extreme values of |z| occur when z lies on the line connecting the origin to the center, either on the far side or near side of the disk.
<p><strong>Step 1:</strong> Identify the geometric region. The inequality |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 O to center C. |C| = |3 - 2i| = √(3² + (-2)²) = √(9 + 4) = √13.</p><p><strong>Step 3:</strong> Determine maximum and minimum values of |z|. Since the disk has radius 4 and center at distance √13 from origin:</p><ul><li>Maximum |z| occurs when z is on the ray from O through C, beyond C: |z|<sub>max</sub> = √13 + 4</li><li>Minimum |z| occurs when z is on the ray from O through C, at the near side: |z|<sub>min</sub> = √13 - 4</li></ul><p><strong>Step 4:</strong> Calculate the difference. Difference = (√13 + 4) - (√13 - 4) = √13 + 4 - √13 + 4 = 8.</p><p>∴ Answer: C (which is 8)</p>
Correct Answer: C