Answer
The polar equation is $ r=-16\sin \theta $.
Work Step by Step
Consider the given equation
${{x}^{2}}+{{\left( y+8 \right)}^{2}}=64$ ;
So,
$\begin{align}
& {{x}^{2}}+{{\left( y+8 \right)}^{2}}=64 \\
& {{\left( r\cos \theta \right)}^{2}}+{{\left( r\cos \theta +8 \right)}^{2}}=64 \\
& {{r}^{2}}{{\cos }^{2}}\theta +{{r}^{2}}{{\sin }^{2}}\theta +16r\sin \theta +64=64 \\
& {{r}^{2}}+16r\sin \theta =0
\end{align}$
So,
$\begin{align}
& {{r}^{2}}=-16r\sin \theta \\
& r=-16\sin \theta
\end{align}$
So, the polar equation is given by $ r=-16\sin \theta $.
Hence, the polar equation is $ r=-16\sin \theta $.
