<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 6.</strong> Complex number \(z_1/z_2\) is</p>
Step-by-Step Solution
Key Concept: The condition |z₁ + z₂|² = |z₁|² + |z₂|² implies that z₁ and z₂ are orthogonal in the complex plane, meaning Re(z₁·z̄₂) = 0, which forces z₁/z₂ to be purely imaginary.
<p><strong>Step 1:</strong> Expand |z₁ + z₂|²:</p><p>|z₁ + z₂|² = (z₁ + z₂)(z̄₁ + z̄₂) = |z₁|² + |z₂|² + z₁z̄₂ + z̄₁z₂</p><p><strong>Step 2:</strong> Note that z₁z̄₂ + z̄₁z₂ = 2Re(z₁z̄₂). Given condition:</p><p>|z₁|² + |z₂|² + 2Re(z₁z̄₂) = |z₁|² + |z₂|²</p><p>Therefore: 2Re(z₁z̄₂) = 0 ⟹ Re(z₁z̄₂) = 0</p><p><strong>Step 3:</strong> Let z₁/z₂ = w. Then z₁z̄₂ = w|z₂|², so:</p><p>Re(w|z₂|²) = 0 ⟹ Re(w) = 0</p><p><strong>Step 4:</strong> Since the real part of z₁/z₂ is zero, z₁/z₂ is purely imaginary (of the form ki where k ∈ ℝ, k ≠ 0).</p><p>∴ Answer: B (purely imaginary)</p>
Correct Answer: B