Answer
The standard form is, ${{\left( x-1 \right)}^{2}}+{{\left( y+2 \right)}^{2}}={{3}^{2}}$.
Work Step by Step
The standard form of the circle is,
${{\left( x-{x}' \right)}^{2}}+{{\left( y-{y}' \right)}^{2}}={{r}^{2}}$
To make the equation in standard form, we will proceed by completing the square method as follows:
$\begin{align}
& {{x}^{2}}+{{y}^{2}}-2x+4y=4 \\
& \left( {{x}^{2}}-2\cdot x\cdot 1+{{1}^{2}} \right)+\left( {{y}^{2}}+2\cdot y\cdot 2+{{2}^{2}} \right)-5=4 \\
& {{\left( x-1 \right)}^{2}}+{{\left( y+2 \right)}^{2}}-5=4 \\
& {{\left( x-1 \right)}^{2}}+{{\left( y+2 \right)}^{2}}=9
\end{align}$
Hence, the required standard form of the circle is,
${{\left( x-1 \right)}^{2}}+{{\left( y+2 \right)}^{2}}={{3}^{2}}$.
