Chapter 1, Equations and Graphs - Section 1.2 - Graphs of Equations in Two Variables; Circles - 1.2 Exercises - Page 103: 83

The equation represents a circle $ (x-1)^2+(y^2+2)^2=2$ with center $(1,-2)$ and radius is $2$

$\underline{\textbf{Given equation}}$ $x^2+y^2-2x+4y+1=0$ $\Rightarrow x^2+y^2-2x+4y=-1$ $\Rightarrow x^2-2x+y^2+4y=-1$ $\Rightarrow (x^2-2x)+(y^2+4y)=-1$ $\Rightarrow (x^2-2x+1)+(y^2+4y+2)=-1+1+2$ $\Rightarrow (x-1)^2+(y^2+2)^2=2$ Which is the standard form for the equation of circle with center $(1,-2)$ and radius is $2$