Is it true that if given a right circular cone a plane that intersects the cone not at the vertex and is parallel to its edge will always result in a parabola regardless of the shape of the cone?

Yes, it is true that if given a right circular cone a plane that intersects the cone not at the vertex and is parallel to its edge will always result in a parabola regardless of the shape of the cone. There are four cases... Plane perpendicular to axis: circle

Plane between perpendicular to axis and parallel to edge: ellipse

Plane parallel to edge: parabola

Plane between parallel to edge and parallel to axis: hyperbolehttp://en.wikipedia.org/wiki/Conic_section