Solution
= Solution
For the <translation>, expand the products of <complex numbers> and collect the terms in $w\bar w$, $\bar w$, $w$ and the constant:
$$
a(w+\alpha)(\bar w+\bar\alpha)+b(\bar w+\bar\alpha)+\bar b(w+\alpha)+c=0.
$$
It follows that
$$
\boxed{A=a,\qquad B=a\alpha+b,\qquad C=a|\alpha|^2+b\bar\alpha+\bar b\alpha+c}.
$$
The last coefficient is real because $b\bar\alpha+\bar b\alpha=2\operatorname{Re}(\bar b\alpha)$.