Find the domain of x and y from the intersection line of two surfaces

Question

I have two point cloud and I use scipy.optimize.curve_fit to get the equations representing two surfaces:

z1 = F1(x, y) = (A1 * x ** 2) + (B1 * y ** 2) + (C1 * x * y) + (D1 * x) + (E1 * y) + F1

z2 = F2(x, y) = (A2 * x ** 2) + (B2 * y ** 2) + (C2 * x * y) + (D2 * x) + (E2 * y) + F2

And the equation of their intersection line is:

MY GOAL: Find the domain of x and y

For example: if x0 is in the domain of the intersection line, what is the range of y corresponding to x0?

I have tried some methods with sympy or scipy, but cannot get the domian.