<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|| and recognize that for |Z - 4/Z| = 2, the maximum |Z| occurs when the constraint is tightest. Substitute |Z| = r and use |4/Z| = 4/r to form an inequality.
<p><strong>Step 1:</strong> Let |Z| = r where r > 0. We know |4/Z| = 4/r.</p><p><strong>Step 2:</strong> Apply the reverse triangle inequality: |Z - 4/Z| ≥ ||Z| - |4/Z|| = |r - 4/r|</p><p>Given |Z - 4/Z| = 2, we have: 2 ≥ |r - 4/r|</p><p><strong>Step 3:</strong> This gives us: -2 ≤ r - 4/r ≤ 2</p><p><strong>Step 4:</strong> For maximum |Z|, consider r - 4/r = 2 (when r > 2):</p><p>r² - 2r - 4 = 0</p><p>r = (2 ± √(4 + 16))/2 = (2 ± √20)/2 = (2 ± 2√5)/2 = 1 ± √5</p><p><strong>Step 5:</strong> Since r > 0, we have r = 1 + √5 (the positive solution with r > 2).</p><p><strong>Step 6:</strong> Verify with r - 4/r = 2: (1 + √5) - 4/(1 + √5) = (1 + √5) - (√5 - 1) = 2 ✓</p><p>∴ Maximum value of |Z| = <strong>1 + √5</strong></p>
Correct Answer: B