1. Find the equation of the circle passing through the two intersection points $P, Q$ of the line $l: 2 x+y+4=0$ and the circle $C: x^{2}+y^{2}+2 x-4 y+1=0$, and having the minimum area.
1. The smallest circle is the one with $P Q$ as its diameter. By eliminating $y$ and $x$ from $2 x+y+4=0$ and $x^{2}+y^{2}+2 x-4 y+1=0$ respectively, we get $g(x)=x^{2}+\frac{26}{5} x+\frac{33}{5}=0, h(y)=y^{2}-\frac{12}{5} y+\frac{4}{5}=0$. Therefore, the equation of the required circle is $x^{2}+y^{2}+\frac{26}{5} x-\frac{12}{5} y+\frac{4}{5}=0$.