Answer
The rectangular equation is, ${{\left( x-4 \right)}^{2}}+{{\left( y-1 \right)}^{2}}=17$
Work Step by Step
Using ${{r}^{2}}={{x}^{2}}+{{y}^{2}}$ we will convert the polar equation to the rectangular equation.
Therefore,
$\begin{align}
& r=8\cos \theta +2\sin \theta \\
& {{r}^{2}}=8r\cos \theta +2r\sin \theta \\
& {{x}^{2}}+{{y}^{2}}=8x+2y
\end{align}$
We can further simplify it by completing the square on x and y as,
$\begin{align}
& {{x}^{2}}+{{y}^{2}}=8x+2y \\
& {{x}^{2}}+{{y}^{2}}-8x-2y+16+1=17 \\
& {{\left( x-4 \right)}^{2}}+{{\left( y-1 \right)}^{2}}=17
\end{align}$
