The brachistochrone is a cycloid that can be given in parametric form by
$$\begin{aligned}x(\phi) &= r(\phi - \sin \phi)\\ y(\phi) &= r(1 - \cos \phi),\end{aligned}$$
where $r$ is the radius of the circle that traces the cycloid. Thus, in order to find $r$ such that the cycloid will connect some point $(x_0,y_0)$ to the origin within the first revolution, we need to solve:
$$ r = \frac{x_0}{\phi-\sin\phi} = \frac{y_0}{1-\cos\phi},\qquad \phi\in [0,2\pi].$$
Rearranging gives the following trigonometric equation:
$$x_0(1-\cos \phi) = y_0(\phi-\sin\phi).$$
How do we solve this equation for any positive $x_0, y_0$?