$\left|z-z_0\right|=2 r $
$\left|\alpha-z_0\right|=r $
$\left|\frac{1}{\bar{\alpha}}-z_0\right|=2 r \quad \alpha \bar{\alpha}=|\alpha|^2 $
$\left|\frac{\alpha}{|\alpha|^2}-z_0\right|=2 r $
$\left(\alpha-z_0\right)\left(\bar{\alpha}-\bar{z}_0\right)=r^2 \Rightarrow|\alpha|^2-z_0\bar{\alpha}-\alpha \bar{z}_0+\left|z_0\right|^2=r^2 $
$\left(\frac{\alpha}{|\alpha|^2}-z_0\right)\left(\frac{\bar{\alpha}}{|\alpha|^2}-\bar{z}_0\right)=4 r^2 \Rightarrow \quad \frac{|\alpha|^2}{|\alpha|^4}-\frac{z_0 \bar{\alpha}}{|\alpha|^2}-\frac{\bar{z}_0 \alpha}{|\alpha|^2}+\left|z_0\right|^2=4 r^2 $
$1-z_0 \bar{\alpha}-\bar{z}_0 \alpha+\left|z_0\right|^2|\alpha|^2=4 r^2|\alpha|^2 $
$\Rightarrow\left(\left|\alpha^2\right|-1\right)+\left|z_0\right|^2\left(1-|\alpha|^2\right)=r^2\left(1-4 \alpha^2\right)$
$\left(|\alpha|^2-1\right)\left(1-\frac{r^2+2}{2}\right)=r^2\left(1-4|\alpha|^2\right) $
$\left(|\alpha|^2-1\right)\left(\frac{-r^2}{2}\right)=r^2\left(1-4|\alpha|^2\right) $
$|\alpha|^2-1=-2+8|\alpha|^2 $
$1=7|\alpha|^2 \quad \Rightarrow \quad|\alpha|=\frac{1}{\sqrt{7}}$
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