Question

$$R^{2}=x^{2}+y^{2}; R=\sqrt{x^{2}+y^{2}}$$

Solve for x, y (complex solution)

$x=\sqrt{R^{2}-y^{2}}\text{, }y\in \mathrm{C}$
$x=-\sqrt{R^{2}-y^{2}}\text{, }y\in \mathrm{C}\text{, }arg(R)<\pi \text{ or }R=0$

Solve for x, y

$x=-\sqrt{R^{2}-y^{2}}\text{, }y\in \begin{bmatrix}-|R|,|R|\end{bmatrix}\text{; }x=\sqrt{R^{2}-y^{2}}\text{, }y\in \begin{bmatrix}-|R|,|R|\end{bmatrix}\text{, }R\geq 0$