Dandelin's Theorem/Directrices

Theorem
Let $\CC$ be a double napped right circular cone with apex $O$.

Let $\PP$ be a plane which intersects $\CC$ such that:
 * $\PP$ does not pass through $O$
 * $\PP$ is not perpendicular to the axis of $\CC$.

Let $\EE$ be the conic section arising as the intersection between $\PP$ and $\CC$.

Let $\SS$ and $\SS'$ be the Dandelin spheres with respect to $\PP$.

Let $\KK$ and $\KK'$ be the planes in which the ring-contacts of $\CC$ with $\SS$ and $\SS'$ are embedded respectively.


 * The intersections of $\KK$ and $\KK'$ with $\PP$ form the directrices of $\EE$.

Also see

 * Definition:Dandelin Spheres