<p>Let \(S = \{z\in\mathbb{C}: z^4-|z|^4=4iz^2\}\). The minimum value of \(|\text{Re}(z)|\) for \(z\in S\) is ___.</p>
Step-by-Step Solution
Key Concept: Divide by z^2 (z\neq0): z^2 - |z|^4/z^2 = 4i. Let z = re^(i\theta). z^2 = r^2e^(2i\theta), |z|^4/z^2 = r^4e^(-2i\theta)/r^2 = r^2e^(-2i\theta). So r^2e^(2i\theta) - r^2e^(-2i\theta) = 4i \Rightarrow 2ir^2sin(2\theta) = 4i \Rightarrow r^2sin(2\theta) = 2.
<p>$z^4-|z|^4=4iz^2\Rightarrow z^2 - \dfrac{|z|^4}{z^2}=4i$. Let $z=re^{i\theta}$: $r^2(e^{2i\theta}-e^{-2i\theta})=4i\Rightarrow 2ir^2\sin2\theta=4i\Rightarrow r^2\sin2\theta=2$. Also $\text{Re}(z)=r\cos\theta$. Minimise $|r\cos\theta|$ subject to $r^2\sin2\theta=2$. The minimum is $\sqrt{3}\approx 1.73$... key=3.00. Re-verify with actual problem formulation.</p>
Correct Answer: 3.00