Solve frac{sqrt{(x-2)^2+y^2}}{sqrt{x^2+y^2}}=sqrt{2}

$$\frac { \sqrt { ( x - 2 ) ^ { 2 } + y ^ { 2 } } } { \sqrt { x ^ { 2 } + y ^ { 2 } } } = \sqrt { 2 }$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=-\sqrt{8-y^{2}}-2\text{, }&\text{unconditionally}\\x=\sqrt{8-y^{2}}-2\text{, }&y\neq -i\text{ and }y\neq i\end{matrix}\right.$$

Solve for y (complex solution)

$$y=-\sqrt{4-4x-x^{2}}$$

$$y=\sqrt{4-4x-x^{2}}\text{, }x\neq 1$$

Solve for x

$$x=-\sqrt{8-y^{2}}-2$$

$$x=\sqrt{8-y^{2}}-2\text{, }|y|\leq 2\sqrt{2}$$

Solve for y

$$y=\sqrt{4-4x-x^{2}}$$

$$y=-\sqrt{4-4x-x^{2}}\text{, }x\geq -2\sqrt{2}-2\text{ and }x\leq 2\sqrt{2}-2$$