Do you have equations for the cone and plane? Getting them on a coordinate axis system would be a good start. If nothing else, it will capacitate an algebraic solution.
equation for the cone is ax^2+bz^2=cy^2
equation for the plane is z=h
the cone is defined by the variables a,b and c.
the plane intersects the cone z=h
I'm looking for an equation which defines the hyperbolas which are the points where the cone intersects the plane.