Equivalence of Definitions of Conic Section

Theorem
The following definitions of a conic section are equivalent:

Proof

 * ApolloniusPappusEquivalence.png

Consider the cone in which $\alpha$ is half the opening angle.

Let the tilting angle of the slicing plane to the horizontal be $\beta$.

The conic section thus defined is given here as an ellipse, but the argument holds for the other cases also.

Hence in the above diagram:
 * $\beta = \angle PDQ$