<p><b>For Problems 26–28:</b> Complex numbers \(z\) satisfy the equation \(|z - (4/z)| = 2\).</p><p>The value of \(\arg(z_1/z_2)\), where \(z_1\) and \(z_2\) are complex numbers with the greatest and the least moduli, can be</p>
Step-by-Step Solution
Key Concept: The equation |z - 4/z| = 2 describes a locus of points in the complex plane. To find extremal moduli, parameterize z = re^(iθ) and minimize/maximize |z| by analyzing the constraint geometrically or algebraically.
<p><strong>Step 1:</strong> Let z = re^(iθ). Then |z - 4/z|² = (z - 4/z)(z̄ - 4/z̄) = |z|² + 16/|z|² - 8cos(2θ) = r² + 16/r² - 8cos(2θ) = 4.</p><p><strong>Step 2:</strong> This gives r² + 16/r² = 4 + 8cos(2θ). For extremal |z| = r, take ∂/∂θ = 0, which occurs when cos(2θ) = ±1. Maximum at cos(2θ) = 1 (θ = 0): r² + 16/r² = 12, so (r² - 6)² = 20, giving r₁ = √(6 + 2√5). Minimum at cos(2θ) = -1 (θ = π): r² + 16/r² = -4 (impossible for real r), so minimum occurs at θ = π/2 or 3π/2.</p><p><strong>Step 3:</strong> At θ = π/2: |z|² = 2 ± 2√2. The extremal moduli satisfy |z₁| · |z₂| = 4 (from the constraint). If z₁ and z₂ are at θ₁ = 0 and θ₂ = π respectively, then arg(z₁/z₂) = 0 - π = -π (or equivalently π).</p><p><strong>Step 4:</strong> Alternatively, z₁ could be at θ = π/2 and z₂ at θ = -π/2, giving arg(z₁/z₂) = π/2 - (-π/2) = π.</p><p>∴ Answer: <strong>B (π or -π, depending on configuration)</strong></p>
Correct Answer: B