Find all couple $(x,y)$ for satisfy $\frac{x+iy}{x-iy}=(x-iy)$ I have a problem to solve this exercise, I hope someone help me.
Find all couple $(x,y)$ for satisfy $\frac{x+iy}{x-iy}=(x-iy)$

$$\frac{z}{\bar{z}} = \bar{z}$$
Take a look at $|\cdot|$ of the two sides and you get $|z| = 1 (=x^2+y^2)$ for free.
Now expand with $z$ (since $z \neq 0$):
$$z^2 = \bar{z}$$
Now consider $\Re z^2 = x^2 - y^2$ and $\Re \bar z = x$ to get
$$x^2 - x =y^2$$
so
$$2x^2 - x = 1$$
which has solutions $x = \pm 1$ implying $y = 0$
