Reduce the number of variables by replacing y with +/- sqrt(1-x^2) . Run twice, once with y being the positive root, and once with y being the negative root, and choose the best of the two.
If K and F are independent of x and y, then you can use calculus: differentiate with respect to x and solve for the zeros.
The roots for the + and - branch come out nearly exactly the same. Both of the branches have roots
(-F +/- sqrt(-F^2 + 2*K^2))/(2*K)
The + branch has root -1/sqrt(2) and the - branch has root +1/sqrt(2)
You would want to verify that these roots are indeed minima and not maxima.