From the previous post (it is now the next one !) it can be seen that projection of the unit circle from the double cone vertex onto the slicing plane will give a second degree equation. What a nice lead in to these equations, and also projective geometry, and to 3D xyz stuff.
The focus/directrix definition is so “rabbit out of a hat” that it’s time for its retirement. In any case, what use is the focus except for a parabola. And the directrix ? Surely “Directrix” is a female director !!!!!!!!!!!!