Getting x in terms of y I have this equation:
$$\dfrac{x}{y} = \dfrac{y-x}{x}$$
How would I separate $x$ and $y$ in $x^2+xy-y^2=0$ ?

Consider your equation, as a quadratic equation with respect to the variable $y$ :
$$-y^2 + xy + x^2$$
handling the variable $x$ as a parameter. 
Then, it would be : 
$$D=b^2-4ac=x^2+4x^2=5x^2$$
It's easy to check that $\forall x\in \mathbb R, D\geq 0$, so you can safely express $y$ with respect to $x$ via the solution of the quadratic equation, without worrying about complex numbers.
Then :
$$y_{1,2} = \frac{-b \pm \sqrt D}{2a} \Rightarrow y_{1,2} = \frac{-x+\sqrt{5x^2}}{-2}$$
