<p>Suppose that <i>z</i> is a complex number that satisfies \(|z - 2 - 2i| \leq 1\). The maximum value of \(|2iz + 4|\) is equal to ___.</p>
Step-by-Step Solution
Key Concept: The constraint |z - 2 - 2i| ≤ 1 describes a closed disk centered at (2, 2i) with radius 1. To maximize |2iz + 4|, we need to find the point z in this disk that is farthest from the point -4/(2i) = 2i in the complex plane, then scale by |2i| = 2.
<p><strong>Step 1:</strong> The constraint |z - 2 - 2i| ≤ 1 describes a closed disk with center C = 2 + 2i and radius r = 1.</p><p><strong>Step 2:</strong> We want to maximize |2iz + 4| = |2(iz + 2)| = 2|iz + 2|. This is equivalent to maximizing |iz + 2|.</p><p><strong>Step 3:</strong> Note that iz + 2 = i(z + 2i) (since iz + 2i·i = iz - 2, we rewrite as iz + 2 = i(z + 2i) requires care). Instead, let w = iz. Since |z - (2 + 2i)| ≤ 1, we have |w/i - (2 + 2i)| ≤ 1, so |-i(w/i) - (2 + 2i)| ≤ 1, giving |w - i(2 + 2i)| = |w - (2i - 2)| ≤ 1.</p><p><strong>Step 4:</strong> Thus w lies in a disk centered at 2i - 2 = -2 + 2i with radius 1. We maximize |w + 2| = |iz + 2|.</p><p><strong>Step 5:</strong> The distance from the center -2 + 2i to the point -2 is |(-2 + 2i) - (-2)| = |2i| = 2. The maximum value of |w + 2| is this distance plus the radius: 2 + 1 = 3.</p><p><strong>Step 6:</strong> Therefore, max|2iz + 4| = 2 · max|iz + 2| = 2 · 3 = 6.</p><p>∴ Answer: <strong>6</strong></p>
Correct Answer: 6