Answer
The required equation in the polar form is $r=-2\,\cos \,\theta $.
Work Step by Step
It is known that $x=r\,\cos \,\theta \ \text{ and }\ y=r\,\sin \,\theta $.
Put $x=r\,\cos \,\theta \ \text{ and }\ y=r\,\sin \,\theta $ in the provided equation ${{\left( x+1 \right)}^{2}}+{{y}^{2}}=1.$
Therefore,
$\begin{align}
& {{\left( r\,\cos \,\theta +1 \right)}^{2}}+{{r}^{2}}\,{{\sin }^{2}}\,\theta =1 \\
& {{r}^{2}}\,{{\cos }^{2}}\,\theta +1+2r\,\cos \,\theta +{{r}^{2}}\,{{\sin }^{2}}\,\theta =1 \\
& {{r}^{2}}\left( {{\cos }^{2}}\,\theta +{{\sin }^{2}}\,\theta \right)+1+2r\,\cos \,\theta =1 \\
& {{r}^{2}}\left( {{\cos }^{2}}\,\theta +{{\sin }^{2}}\,\theta \right)+2r\,\cos \,\theta =0
\end{align}$
and
${{r}^{2}}=-2r\,\cos \,\theta $
This implies that $r=-2\,\cos \,\theta $.
