<p>If \( |z| = \left| z - \dfrac{4}{z} \right| + \dfrac{4}{|z|} \), then the maximum value of \( |z| \) is</p>
<p>\( \sqrt{5} + 1 \)</p>
<p>\( \sqrt{5} - 1 \)</p>
<p>\( 1 + \sqrt{5} \)</p>
<p>\( 2 \)</p>
Step-by-Step Solution
Key Concept: Rearrange the equation as |z| - 4/|z| = |z - 4/z| and recognize that for complex z, the RHS represents a geometric constraint. The maximum occurs when z is real and positive, transforming this into an algebraic inequality in |z|.
<p><strong>Step 1:</strong> Let |z| = r where r > 0. Rewrite the given equation:</p><p>r = |z - 4/z| + 4/r</p><p>∴ r - 4/r = |z - 4/z|</p><p><strong>Step 2:</strong> By the triangle inequality, |z - 4/z| ≤ |z| + 4/|z| = r + 4/r.</p><p>But we also need |z - 4/z| ≥ ||z| - 4/|z|| = |r - 4/r|.</p><p>From our equation: |z - 4/z| = r - 4/r (assuming r > 4/r, so r² > 4).</p><p><strong>Step 3:</strong> For this to hold, we need z to be real and positive (z = r). Then:</p><p>|r - 4/r| = r - 4/r, which requires r ≥ 2.</p><p><strong>Step 4:</strong> Substitute z = r (real) into the original equation:</p><p>r = |r - 4/r| + 4/r</p><p>r = r - 4/r + 4/r = r ✓</p><p>This is satisfied for all r ≥ 2. However, checking the constraint more carefully:</p><p>When z = r > 0: |r - 4/r| + 4/r must equal r.</p><p><strong>Step 5:</strong> By AM-GM inequality applied to r + 4/r ≥ 2√(r·4/r) = 4, the minimum value of r + 4/r is 4 (when r = 2).</p><p>For the equation r = |r - 4/r| + 4/r to have maximum r, we solve: if r > 2, then r - 4/r = r - 4/r, giving equality only at boundary.</p><p>Testing r = 2√2: |2√2 - 4/(2√2)| + 4/(2√2) = |2√2 - √2| + √2 = √2 + √2 = 2√2 ✓</p><p><strong>∴ Answer: Maximum value of |z| = 2√2</strong></p>
Correct Answer: A