To be honest I am hoping that there is an obvious solution to this problem that I simply missed. I actually wrote a small program that numerically solves this problem, it is indeed very easy to find a solution. If you look at it graphically it's obvious how to get there. However the final application that this problem is a part of is quite a lot more involved, that's why an analytical solution would be very beneficial.

True, thanks, I edited the question to make it clearer.