<p>If \(\left|z - \dfrac{4}{z}\right| = 2\), then the maximum value of \(|z|\) is equal to</p>
Step-by-Step Solution
Key Concept: Use the reverse triangle inequality |a - b| ≥ ||a| - |b|| combined with algebraic manipulation. Set |z| = r and recognize that the constraint |z - 4/z| = 2 creates bounds on r by analyzing when equality conditions are satisfied.
<p><strong>Step 1:</strong> Let |z| = r where r > 0. Use the reverse triangle inequality:</p><p>||z| - |4/z|| ≤ |z - 4/z| = 2</p><p>This gives: |r - 4/r| ≤ 2</p><p><strong>Step 2:</strong> For maximum r, consider when r ≥ 4/r (i.e., r² ≥ 4, so r ≥ 2):</p><p>r - 4/r ≤ 2</p><p>r² - 2r - 4 ≤ 0</p><p><strong>Step 3:</strong> Solve r² - 2r - 4 = 0:</p><p>r = (2 ± √(4 + 16))/2 = (2 ± √20)/2 = (2 ± 2√5)/2 = 1 ± √5</p><p>Since r > 0 and r ≥ 2, we take r = 1 + √5 ≈ 3.236</p><p><strong>Step 4:</strong> Verify: For r = 1 + √5, when z and 4/z point oppositely, |z - 4/z| = r - 4/r = (1 + √5) - 4/(1 + √5) = (1 + √5) - (√5 - 1) = 2 ✓</p><p>∴ Maximum value of |z| = <strong>1 + √5</strong></p>
Correct Answer: B