<p>If \(z \leq 4\), then find the maximum value of \(|iz + 3 - 4i|\).</p>
Step-by-Step Solution
Key Concept: Rewrite |iz + 3 - 4i| as |i(z - (4 + 3i))| = |z - (4 + 3i)| to recognize this as the distance from z to the fixed point (4 + 3i). The maximum distance occurs when z lies on the circle |z| ≤ 4 at the point farthest from (4 + 3i).
<p><strong>Step 1:</strong> Rewrite the expression using |i| = 1:</p><p>|iz + 3 - 4i| = |i(z - (4 + 3i))| = |i| · |z - (4 + 3i)| = |z - (4 + 3i)|</p><p><strong>Step 2:</strong> Recognize that |z - (4 + 3i)| represents the distance from z to the point w₀ = 4 + 3i in the complex plane.</p><p><strong>Step 3:</strong> Calculate the distance from origin to w₀:</p><p>|w₀| = |4 + 3i| = √(16 + 9) = √25 = 5</p><p><strong>Step 4:</strong> Since z must satisfy |z| ≤ 4 (z lies within or on a circle of radius 4 centered at origin), the maximum distance from z to w₀ occurs when z is on the boundary circle |z| = 4, positioned directly away from w₀ through the origin.</p><p><strong>Step 5:</strong> The maximum distance is:</p><p>max|z - w₀| = |w₀| + radius = 5 + 4 = 9</p><p>∴ Answer: <strong>9</strong></p>
Correct Answer: 9