<p><b>For Problems 26–28:</b> Complex numbers \(z\) satisfy the equation \(|z - (4/z)| = 2\).</p><p>The difference between the least and the greatest moduli of complex numbers is</p>
Step-by-Step Solution
Key Concept: Rewrite the equation as |z² - 4| = 2|z| and use the substitution |z| = r to convert the complex constraint into a real inequality. This geometric condition represents an annulus (ring), and extremal |z| values occur when z is real.
<p><strong>Step 1:</strong> Multiply the equation |z - 4/z| = 2 by |z| (assuming z ≠ 0):</p><p>|z² - 4| = 2|z|</p><p><strong>Step 2:</strong> Let |z| = r where r > 0. For z on the real axis (where extrema occur), let z = x (real):</p><p>|x² - 4| = 2|x|</p><p><strong>Step 3:</strong> Case 1 (x² ≥ 4, so |x| ≥ 2):</p><p>x² - 4 = 2|x|</p><p>x² - 2|x| - 4 = 0</p><p>|x| = (2 ± √(4 + 16))/2 = (2 ± √20)/2 = 1 ± √5</p><p>Since |x| ≥ 2: |x| = 1 + √5 ≈ 3.236</p><p><strong>Step 4:</strong> Case 2 (x² < 4, so |x| < 2):</p><p>4 - x² = 2|x|</p><p>x² + 2|x| - 4 = 0</p><p>|x| = (-2 ± √(4 + 16))/2 = (-2 ± √20)/2</p><p>Since |x| > 0: |x| = -1 + √5 ≈ 1.236</p><p><strong>Step 5:</strong> The difference between greatest and least moduli:</p><p>(1 + √5) - (-1 + √5) = 2</p><p>∴ Answer: A</p>
Correct Answer: A