Given the function of a double cone and a plane, how do we find the intersection between them?
Suppose the equation of the cone is $f(x, y, z) = 0$ and the equation of the plane is $h(x, y, z) = 0$. Would the intersection be given by $f(x, y, z) = h(x, y, z)$?
It would be given by $f(x, y, z) = h(x, y, z) = 0.$ You can also work out the kind of section, by determining how many lines in the cone are parallel to the plane: 0 for an ellipse, 1 for a parabola and 2 for a hyperbola.