<p>Modulus of nonzero complex number <i>z</i> satisfying \(\bar{z} + z = 0\) and \(|z|^2 - 4zi = z^2\) is ___.</p>
Step-by-Step Solution
Key Concept: From z̄ + z = 0, we get z is purely imaginary (z = bi). Substitute into |z|² - 4zi = z² to find the modulus.
<p><strong>Step 1:</strong> From condition z̄ + z = 0:<br>If z = a + bi, then (a - bi) + (a + bi) = 0<br>This gives 2a = 0, so a = 0<br>Therefore z = bi (purely imaginary, b ≠ 0)</p><p><strong>Step 2:</strong> Substitute z = bi into |z|² - 4zi = z²:<br>|bi|² - 4(bi)i = (bi)²<br>b² - 4bi² = b²i²<br>b² - 4b(-1) = -b²<br>b² + 4b = -b²<br>2b² + 4b = 0<br>2b(b + 2) = 0<br>Since z is nonzero, b ≠ 0, so b = -2</p><p><strong>Step 3:</strong> Therefore z = -2i<br>|z| = |-2i| = 2</p><p>∴ Answer: <strong>2</strong></p>
Correct Answer: 2