What is the cone of the conic section? Given the general (real valued) equation of a
conic section:
$$
  A x^2 + B xy + C y^2 + D x + E y + F = 0
$$
Then what is the circular cone associated with it ? Is it unique ?
And is there a way to derive its equation, expressed in $(A,B,C,D,E,F)$ and $(x,y,z)$ ?
I've done some homework
here and
here,
but wasn't able to extract a simple method to find the cone(s), given the conic section.

@Jyrki's suggestion to consider Dandelin Spheres is the key. It's possible (even easy) to construct a family of Dandelin Spheres from a particular conic, and these give the family of cones you seek.

Let's take the case of an ellipse. Viewing the curve's plane edge-on, we visually collapse the ellipse to its major axis $\overline{PQ}$. Let $F$ and $F^\prime$ be the foci.
Choose any $R$ such that $\overline{RF}\perp\overline{PQ}$, and draw the circle through $F$ with center $R$. This circle is the intersection of a Dandelin sphere with the plane perpendicular to the ellipse through its major axis.

Draw circles about $P$ and $Q$ through $F$ to determine points $S$ and $T$ on $\bigcirc R$. Necessarily, $\overleftrightarrow{PS}$ and $\overleftrightarrow{QT}$ are tangent to the circle. Let $C$ be the point where these lines meet.

The incircle of $\triangle PQC$ is our other "Dandelin circle".

From here, we see that $C$ is the apex the of the cone we seek. (If the tangent lines are parallel, then $C$ is the "point at infinity" and our cone is actually a cylinder.) Those tangent lines are the intersections of the cone with the perpendicular plane. Thus, we get a family of such cones based on the parameter-point $R$.
Finding the equation of the cone family should be relatively straightforward for an ellipse in "standard position". For a general ellipse, a few coordinate transformations will be helpful. We handle parabolas and hyperbolas similarly.
