Deriving Descriptions of Conic Sections from Fundamental Definition

[CarlisleLes]

Everyone knows by now that a conic section is the figure formed when a plane intersects a right circular cone. Most everyone also knows that there are many different ways to describe a conic, geometrically and algebraically. What one seldom sees is the derivation of those descriptions from the fundamental definition. Using Dandelin Spheres it is easy to accomplish this for an ellipse. What I have never seen is a proof, based on the fundamental definition, of the equivalence of the ratio of the distances of a point on the conic to the focus and to the directrix, or even a definition of the directrix itself. Can anyone supply or direct me to such information? Thanks.

 

[mathman]

http://en.wikipedia.org/wiki/Conic_section

Try the above.

 

[CarlisleLes]

I have read that a dozen times. It doesn't explain the directrix derivation - just says it's possible to do so.

 

[mathman]

http://mathworld.wolfram.com/ConicSectionDirectrix.html

This has derivations.

 

[CarlisleLes]

I found an answer at http://www.clowder.net/hop/Dandelin/Dandelin.html. Thanks everyone.