A degeneate is a the empty set (no intersection which seems impossible), a pairs of intersecting lines or a point. The way you get a pair of intersection lines is by passing a plane perpendicular to the cone base, but that is a type of hyperbola shape. So that point is the only thing you can get, by passing a plane parrallel with with the side of the cone through the vertex of the double cone.

This is mine 33th Post!!!