Answer
Equation of a circle with center $\left( -3,5 \right)$ and radius $4$ is ${{\left( x+3 \right)}^{2}}+{{\left( y-5 \right)}^{2}}=16$
Work Step by Step
First we use the formula of circumference of a circle to calculate the radius, $C=2\pi r$.
Put the value of circumference $8\pi $ in formula $C=2\pi r$.
$\begin{align}
& 2\pi r=8\pi \\
& r=\frac{8\pi }{2\pi } \\
& r=4 \\
\end{align}$
Put the value of radius $4$ and center coordinate $\left( -3,5 \right)$ in the standard equation of the circle:
$\begin{align}
& {{\left( x-h \right)}^{2}}+{{\left( y-k \right)}^{2}}={{r}^{2}} \\
& {{\left( x-\left( -3 \right) \right)}^{2}}+{{\left( y-\left( 5 \right) \right)}^{2}}={{\left( 4 \right)}^{2}} \\
& {{\left( x+3 \right)}^{2}}+{{\left( y-5 \right)}^{2}}=16
\end{align}$
Thus, the equation of the circle with center $\left( -3,5 \right)$ and radius $4$ is ${{\left( x+3 \right)}^{2}}+{{\left( y-5 \right)}^{2}}=16$.
