Optimizing an ellipse

Question

I wish to find the lengths of the major and minor axis of an ellipse obtained by a plane intersected an ellipsoid, but I wish to do this generally. So far I know that the definition of a plane is $$Ax + By + Cz = 0$$ and the definition of an ellipsoid is $$\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2 = 1$$ My approach is to use Lagrange Multpliers and try to maximize the square of the distance function. Altogether that is the following function: $$F(x,y,z) = x^2 + y^2 + z^2 + \lambda(Ax + By + Cz) + \mu(\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 + \left(\frac{z}{c}\right)^2-1$$ The algebra is getting quite messy in solving the resulting nonlinear system which leads me to believe I am doing something wrong or that there is a simpler approach. Does anyone have any idea?