Cone and vertical plane are given.
Draw angle bisectors of plane and cone surface.
Their intersections with the center line of the cone give the centers of the Dandelin spheres.
The plane normals through these points lead to the focal points of the ellipse.

a = major radius
b = minor radius
e = linear eccentricity = distance from center to focus

Using the Pythagorean theorem b can now be deduced from a and e.

Image set Shed with conic dormer
Technical drawings of a house with a shed roof and a conic dormer. The intersection of the cone with the plane of the dormer window in an ellipse. That with the roof is a hyperbola.