2. Find the equation of the circle passing through the intersection points of the two circles
$$
x^{2}+y^{2}+6 x-4=0 \text { and } x^{2}+y^{2}+6 y-28=0
$$

and whose center lies on the line $x-y-4=0$.
(Tip: Let the equation of the required circle be $x^{2}+y^{2}+6 x-4+$ $\lambda\left(x^{2}+y^{2}+6 x-28\right)=0$. Convert it to the standard form of a circle and substitute the coordinates of the center into the given line equation to find $\lambda=-7$. The equation of the required circle is $\left(x-\frac{1}{2}\right)^{2}+\left(y+\frac{7}{2}\right)^{2}=\frac{178}{4}$. )