We know that \(\displaystyle{r} \cos{\theta}={x}\) and that \(\displaystyle{r} \sin{\theta}={y}\), so \(\displaystyle\frac{x}{{r}}= \cos{\theta}{\quad\text{and}\quad}\frac{y}{{r}}.\)

Thus, \(\displaystyle{r}=\frac{y}{{r}}–\frac{x}{{r}}.\)

Multiplying both sides by r gives:

\(\displaystyle{R}^{2}={y}–{x}.\)

We know that \(\displaystyle{r}=\sqrt{{{x}^{2}+{y}^{2}}}\), then

\(\displaystyle{x}^{2}+{y}^{2}={y}-{x}\)