## Task B-4.7.

Determine all complex numbers $z=x+yi, x, y \in \mathbb{R}$ such that the point $(x, y)$ is in the fourth quadrant and the following conditions are satisfied:

$$
|z+iz|=2, \quad \operatorname{Re}\left(z^{4}\right)=-2
$$
## First solution.

Let $z=x+yi, x>0, y<0$ :

$$
z_{1}=\frac{\sqrt{2}}{2}-i \frac{\sqrt{6}}{2}, \quad z_{2}=\frac{\sqrt{6}}{2}-i \frac{\sqrt{2}}{2}
$$