Answer
$$r=2 c \cos \theta$$
Work Step by Step
Since
\begin{align*}
x^{2}+y^{2}&=2 c x \\
r^{2}&=2 c r \cos \theta \\
r^{2}-2 c r \cos \theta&=0 \\
r(r-2 c \cos \theta)&=0
\end{align*}
Then $ r=0$ or $r=2 c \cos \theta$
Since for $\theta=\frac{\pi}{2}+n \pi,$ $r=0$, then the curve is represented by
$$r=2 c \cos \theta$$