<p>If <em>z</em> is any complex number satisfying \(|z - 3 - 2i| \leq 2\), then the minimum value of \(|2z - 6 + 5i|\) is _______.</p><p>(IIT-JEE, 2011)</p>
Step-by-Step Solution
Key Concept: The constraint |z - 3 - 2i| ≤ 2 describes a closed disk in the complex plane centered at (3, 2) with radius 2. To minimize |2z - 6 + 5i|, rewrite it as 2|z - 3 + (5i/2)| and find the distance from the disk's center to the target point, then subtract the radius.
<p><strong>Step 1:</strong> Rewrite the expression to minimize.</p><p>|2z - 6 + 5i| = |2(z - 3) + 5i| = |2(z - 3 + 5i/2)|= 2|z - 3 + 5i/2|</p><p><strong>Step 2:</strong> Identify the constraint region.</p><p>The constraint |z - 3 - 2i| ≤ 2 represents a closed disk centered at C = 3 + 2i with radius r = 2.</p><p><strong>Step 3:</strong> Rewrite the target point.</p><p>We need to minimize 2|z - (3 - 5i/2)|, which means finding the minimum distance from z to the point P = 3 - 5i/2 (or 3 - 2.5i).</p><p><strong>Step 4:</strong> Calculate distance from disk center to target point.</p><p>Distance = |C - P| = |(3 + 2i) - (3 - 5i/2)| = |2i + 5i/2| = |9i/2| = 9/2</p><p><strong>Step 5:</strong> Apply geometric principle.</p><p>Since P lies outside the disk (distance 9/2 > radius 2), the minimum distance from any point z in the disk to P is:</p><p>min|z - P| = 9/2 - 2 = 5/2</p><p><strong>Step 6:</strong> Multiply by the factor of 2.</p><p>Minimum of |2z - 6 + 5i| = 2 × (5/2) = <strong>5</strong></p>
Correct Answer: 5