Answer
$x^{2}+y^{2}=(x^{2}+y^{2}-x)^{2}$
Work Step by Step
Formulas relating rectangular $(x,y)$ and polar $(r,\theta)$ coordinates:
$ x=r\cos\theta$ and $ y=r\sin\theta$
$r^{2}=x^{2}+y^{2}$ and $\displaystyle \tan\theta=\frac{y}{x}$.
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$ r=1+\cos\theta$
... multiply both sides with r,
$ r^{2}=r+r\cos\theta$
... substitute $r^{2}$ and $ r\cos\theta$ using the formulas,
$x^{2}+y^{2}=r+x\qquad /-x$
$ r=x^{2}+y^{2}-x\qquad$... square both sides,
$r^{2}=(x^{2}+y^{2}-x)^{2}\qquad $... substitute $r^{2}$
$x^{2}+y^{2}=(x^{2}+y^{2}-x)^{2},$
...we have an equation in rectangular coordinates.