Answer
$r=4\cos \theta $.
Work Step by Step
Rectangular equation is ${{\left( x-2 \right)}^{2}}+{{y}^{2}}=4$ …… (I)
The relation between polar coordinates and rectangular coordinates can be represented as below:
$x=r\cos \theta \ \text{ and }\ y=r\sin \theta $ …… (II)
Substitute values of $x\ \text{ and }\ y$ from (II) in (I) to get
$\begin{align}
& {{\left( x-2 \right)}^{2}}+{{y}^{2}}=4 \\
& {{\left( r\cos \theta -2 \right)}^{2}}+{{\left( r\sin \theta \right)}^{2}}=4 \\
& {{r}^{2}}{{\cos }^{2}}\theta +4-4r\cos \theta +{{r}^{2}}{{\sin }^{2}}\theta =4 \\
& {{r}^{2}}\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)-4r\cos \theta =0
\end{align}$
As,
$\left( {{\cos }^{2}}\theta +{{\sin }^{2}}\theta \right)=1$
From here we get,
$\begin{align}
& {{r}^{2}}-4r\cos \theta =0 \\
& {{r}^{2}}=4r\cos \theta \\
& r=4\cos \theta
\end{align}$
Hence the obtained polar expression is $r=4\cos \theta $.
