<p>If \(|z_1| = 15\) and \(|z_2 - 3 - 4i| = 5\), then</p>
<p>\(|z_1 - z_2|_{\min} = 5\)</p>
<p>\(|z_1 - z_2|_{\min} = 10\)</p>
<p>\(|z_1 - z_2|_{\max} = 20\)</p>
<p>\(|z_1 - z_2|_{\max} = 25\)</p>
Step-by-Step Solution
Key Concept: The condition |z₂ - 3 - 4i| = 5 means z₂ lies on a circle centered at (3, 4) with radius 5. Combined with |z₁| = 15, we must find the range of |z₁ - z₂| by considering extreme positions on this circle relative to the origin.
<p><strong>Step 1:</strong> Interpret the constraints. |z₁| = 15 means z₁ lies on a circle of radius 15 centered at origin. |z₂ - 3 - 4i| = 5 means z₂ lies on a circle of radius 5 centered at point (3, 4).</p><p><strong>Step 2:</strong> The distance from origin to center C = (3, 4) is |C| = √(9 + 16) = 5.</p><p><strong>Step 3:</strong> For |z₁ - z₂|, consider that z₁ is on circle of radius 15 (at origin) and z₂ is on circle of radius 5 (centered at distance 5 from origin).</p><p><strong>Step 4:</strong> Maximum value: When z₁ and z₂ are on opposite sides, |z₁ - z₂|<sub>max</sub> = 15 + 5 + 5 = 25 (z₁ at distance 15 from O, z₂ at distance 5 away from center on far side).</p><p><strong>Step 5:</strong> Minimum value: When z₁ and z₂ are aligned toward center, |z₁ - z₂|<sub>min</sub> = 15 - 5 - 5 = 5 (z₁ closest to center along the ray, z₂ closest to origin on the circle).</p><p><strong>Step 6:</strong> Therefore: 5 ≤ |z₁ - z₂| ≤ 25.</p><p>∴ Answer: BD (assuming B and D represent the range bounds)</p>
Correct Answer: BD