The Theorem of DandelinIn the Main Conics Page, we define a conic in terms of the focus, directrix and eccentricity.Conics may also be defined as plane sections of a right circular cone. In the picture on the right, the pink plane cuts the pale blue cone in a curve C.
To show that the definitions are equivalent,
Imagine a sphere sitting inside the top sheet of the cone.
Using the symmetry of the cone about its axis,
If the cutting (pink) plane is also perpendicular to the axis,
Otherwise, the cutting plane meets the green plane meet in a line D.


The picture below shows the three cases (ellipse, parabola, hyperbola)
A complete proof (in french) with nice graphics appears here. 
Geometry Page  Main Conics Page  Main Cabri Page 