I am pretty confident I understand how to use Variational Calculus in order to find out the path between two points on which a bead could slide under gravity in the shortest time. If starting from $(0,0)$, the path is described by $$x(\phi) = a(\phi-\sin(\phi))$$ $$y(\phi) = a(\cos{\phi}-1)$$
But my problem is when determining the exact fastest path between a point and a given line. For example, if the initial point is $A = (0,0)$ and I want the fastest path from $A$ to line $x=1$, I don't know which brachistochrone curve should I choose. I have tried to use Lagrange multipliers, but I couldn't come to anything meaningful.
Any help would be appreciated!
The answer is $\:\phi_{1}=\pi\:$ and $\:\alpha=1/\pi$.
Minima at : $\phi = (2k+1)\pi,\quad k=0,1,2,3\cdots\:$.
Turning points where : $\tan\left(\dfrac{\phi}{2}\right) = \dfrac{\phi}{2}\:$.