Reconciliation of Cone-Slicing and Focus-Directrix Definitions of Conic Sections

Question

It is well known that the family of conics is derived by slicing an infinite double-napped right circular cone, with the specific type of conic depending on the angle of slice.

Separately it is also know that these conics may be defined by its focus (or focis, as the case may be) and directrix, from which standard equations are derived.

What is not obvious is how these two concept definining the family of conics are to each other.

Question:
Show that the cone-slicing defintion of conic sections is equivalent to the focus-directrix definition. Indicate where the focus and directrix are located in relation to the double-napped cone.

Diagrams would be helpful.

Answer 1

The inner and outer spheres tangent internally to a cone and also to a plane intersecting the cone are called Dandelin spheres. The limit case with one such sphere gives rise to the parabola.

For all cases, the foc(us)(i) is/are the tangent point(s) of the sphere(s) to the cutting plane.

The directri(x)(ces) is/are the intersection line(s) of the plane of the conic (the cutting plane) and the plane(s) of the circle(s) of tangency of the cone and the sphere(s).

The circle (the directrix line is at infinity):

The parabola: