<p>The complex number <i>z</i> satisfies \(z + |z| = 2 + 8i\). The value of \(|z|\) is ___.</p>
Step-by-Step Solution
Key Concept: Separate the complex number z into real and imaginary parts, then use the fact that |z| is purely real to equate real and imaginary components independently.
<p><strong>Step 1:</strong> Let z = x + yi where x, y ∈ ℝ. Then |z| = √(x² + y²), which is real.</p><p><strong>Step 2:</strong> Substitute into z + |z| = 2 + 8i:</p><p>(x + yi) + √(x² + y²) = 2 + 8i</p><p><strong>Step 3:</strong> Separate into real and imaginary parts:</p><p>Real part: x + √(x² + y²) = 2</p><p>Imaginary part: y = 8</p><p><strong>Step 4:</strong> From the imaginary part, y = 8. Substitute into the real part equation:</p><p>x + √(x² + 64) = 2</p><p>√(x² + 64) = 2 - x</p><p><strong>Step 5:</strong> Square both sides (valid when 2 - x ≥ 0, so x ≤ 2):</p><p>x² + 64 = 4 - 4x + x²</p><p>64 = 4 - 4x</p><p>4x = -60</p><p>x = -15</p><p><strong>Step 6:</strong> Verify x = -15 ≤ 2 ✓ and check: √(225 + 64) = √289 = 17, and 2 - (-15) = 17 ✓</p><p><strong>Step 7:</strong> Calculate |z|:</p><p>|z| = √((-15)² + 8²) = √(225 + 64) = √289 = 17</p><p>∴ Answer: <strong>17</strong></p>
Correct Answer: 17