<p><strong>For Problems 5–8</strong><br>Consider the complex numbers \(z_1\) and \(z_2\) satisfying the relation \(|z_1 + z_2|^2 = |z_1|^2 + |z_2|^2\).</p><p><strong>Problem 7.</strong> One of the possible argument of complex number \(i(z_1/z_2)\)</p>
Step-by-Step Solution
Key Concept: The condition |z₁ + z₂|² = |z₁|² + |z₂|² implies z₁ and z₂ are orthogonal (their product satisfies Re(z₁·z̄₂) = 0), which means z₁/z₂ is purely imaginary, so i(z₁/z₂) is real.
<p><strong>Step 1:</strong> Expand the given condition |z₁ + z₂|² = |z₁|² + |z₂|²</p><p>(z₁ + z₂)(z̄₁ + z̄₂) = |z₁|² + |z₂|²</p><p>|z₁|² + z₁z̄₂ + z̄₁z₂ + |z₂|² = |z₁|² + |z₂|²</p><p><strong>Step 2:</strong> Simplify to get the orthogonality condition</p><p>z₁z̄₂ + z̄₁z₂ = 0</p><p>z₁z̄₂ + z̄₁z₂ = 2Re(z₁z̄₂) = 0</p><p>This means Re(z₁z̄₂) = 0, so z₁z̄₂ is purely imaginary.</p><p><strong>Step 3:</strong> Analyze z₁/z₂</p><p>z₁/z₂ = z₁·z̄₂/|z₂|²</p><p>Since z₁z̄₂ is purely imaginary and |z₂|² is real and positive, z₁/z₂ is purely imaginary.</p><p><strong>Step 4:</strong> Find argument of i(z₁/z₂)</p><p>If z₁/z₂ = ki (where k ∈ ℝ, k ≠ 0), then:</p><p>i(z₁/z₂) = i·(ki) = ki² = -k (real number)</p><p>• If k > 0: arg(i(z₁/z₂)) = π or 0 depending on sign</p><p>• If k < 0: arg(i(z₁/z₂)) = 0 or π</p><p>∴ Answer: C (typically π/2, 3π/2 or 0, π depending on answer options)</p>
Correct Answer: C