Let $S=\{z\in\mathbb{C}:\ \bar{z}=i(z^2+\text{Re}(\bar{z}))\}$. Then $\displaystyle\sum_{z\in S}|z|^2$ is equal to
Step-by-Step Solution
Key Concept: Substitute $z=x+iy$, $\bar{z}=x-iy$. Real part condition gives $x(2y+1)=0$, so $x=0$ or $y=-\frac{1}{2}$. Imaginary part gives a second condition.
Four solutions: $0,i,\frac{1-i}{2},-\frac{3+i}{2}$. $\sum|z|^2=0+1+\frac{1}{2}+\frac{5}{2}=4$.
Correct Answer: 2