Starting with the cone...

Contrary to popular belief, a cone has two portions separated from each other by the vertex; it has no base or end and extends infinitely far in both directions.

 

Picturing the conic section...

A conic section is a curve formed by the intersection of a plane with a right circular cone. The traditional conic sections are the parabola, ellipse (of which a circle is a special case), and hyperbola. Conic sections have many intersting and important applications in science, engineering, and industry.

Picture copied from the book Calculus by Douglas F. Riddle, Wadsworth, Inc. 1984.