Example 4. Find the equation of the circle passing through points $A(2,2), B(5,3), C(3$, $-1)$

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
The family of circles passing through point $A$ is given by
$$
\begin{array}{c}
(x-2)^{2}+(y-2)^{2} \\
+\lambda(x-2)+\mu(y-2)=
\end{array}
$$
0. This family of circles also passes through points $B(5,3)$ and $C(3,-1)$, leading to
$$
\left\{\begin{array}{c}
(5-2)^{2}+(3-2)^{2}+\lambda(5-2) \\
+\mu(3-2)=0, \\
(3-2)^{2}+(-1-2)^{2}+\lambda(3-2) \\
+\mu(-1-2)=0 .
\end{array}\right.
$$
$$
\begin{array}{c}
(3-2)^{2}+(-1-2)^{2}+\lambda(3-2) \\
+\mu(-1-2)=0
\end{array}
$$

which simplifies to $\left\{\begin{array}{l}9+1+3 \lambda+\mu=0 \\ 1+9+\lambda-3 \mu=0\end{array}\right.$.
Thus, $\left\{\begin{array}{l}\lambda=-4, \\ \mu=+2\end{array}\right.$
The equation of the required circle is
$$
x^{2}+y^{2}-8 x-2 y+12=0 \text {. }
$$