Inverse of stereographic projection of a circle on a line

Question

I am curious about the stereographic projection of the unit cycle centered at $0$, from the ''north pole'' $(0,1)$ on the line $y = -1$. We know: $$\phi: S_1/ \{(0,1)\} \to \mathbb{R}$$ can be given by $\phi(x,y) = \frac{2x}{1-y}$. I am trying to find its inverse. That is, given $u \in \mathbb{R}$ find $(x,y) \in S_1/ \{(0,1)\}$