Question:

Let $S=\{z\in\mathbb{C}:4z^2+\bar{z}=0\}$. Then $\displaystyle\sum_{z\in S}|z|^2$ is equal to:

$\dfrac{5}{64}$

$\dfrac{1}{16}$

$\dfrac{3}{16}$

$\dfrac{7}{64}$

Step-by-Step Solution

Key Concept: Let $z=x+iy$. Separating real and imaginary parts of $4z^2+\bar{z}=0$: Real: $4x^2-4y^2+x=0$. Imaginary: $y(8x-1)=0$.

4 solutions. $\sum|z|^2=3/16$.

Correct Answer: 3