Solve x^2+varepsilony^2=varepsilon6

$$x ^ { 2 } + \varepsilon y ^ { 2 } = \varepsilon 6$$

Solve for x (complex solution)

$$x=-\sqrt{\epsilon _{6}-\epsilon y^{2}}$$

$$x=\sqrt{\epsilon _{6}-\epsilon y^{2}}$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=-i\epsilon ^{-\frac{1}{2}}\sqrt{x^{2}-\epsilon _{6}}\text{; }y=i\epsilon ^{-\frac{1}{2}}\sqrt{x^{2}-\epsilon _{6}}\text{, }&\epsilon \neq 0\\y\in \mathrm{C}\text{, }&\epsilon _{6}=x^{2}\text{ and }\epsilon =0\end{matrix}\right.$$

Solve for x

$$x=\sqrt{\epsilon _{6}-\epsilon y^{2}}$$

$$x=-\sqrt{\epsilon _{6}-\epsilon y^{2}}\text{, }\epsilon \leq \frac{\epsilon _{6}}{y^{2}}\text{ or }y=0$$

Solve for y

$$\left\{\begin{matrix}y=\sqrt{\frac{\epsilon _{6}-x^{2}}{\epsilon }}\text{; }y=-\sqrt{\frac{\epsilon _{6}-x^{2}}{\epsilon }}\text{, }&\left(|x|\geq \sqrt{\epsilon _{6}}\text{ and }\epsilon <0\right)\text{ or }\left(\epsilon _{6}\geq 0\text{ and }\epsilon \neq 0\text{ and }|x|=\sqrt{\epsilon _{6}}\right)\text{ or }\left(\epsilon _{6}\geq 0\text{ and }|x|\leq \sqrt{\epsilon _{6}}\text{ and }\epsilon >0\right)\text{ or }\left(\epsilon _{6}<0\text{ and }\epsilon <0\right)\\y\in \mathrm{R}\text{, }&\epsilon _{6}=x^{2}\text{ and }\epsilon =0\end{matrix}\right.$$