It is well known that the family of conics is derived by slicing an infinite double-napped right circular cone, with the specific type of conic depending on the angle of slice.
Separately it is also know that these conics may be defined by its focus (or focis, as the case may be) and directrix, from which standard equations are derived.
What is not obvious is how these two concept definining the family of conics are to each other.
Show that the cone-slicing defintion of conic sections is equivalent to the focus-directrix definition. Indicate where the focus and directrix are located in relation to the double-napped cone.
Diagrams would be helpful.