The determinant $\Delta$ for the general equation of a conic $ax^2+2hxy+by^2+2gx+2fy+c=0$ is given by the following:
$$\Delta=\left| \begin{array} {ccc} a & h & g \\ h & b & f \\ g &f &c\\ \end{array} \right|$$
This determinant, when equal to $0$, the equation represents a pair of straight lines. When non-zero, the equation represents a non-degenerate conic (circle, ellipse, parabola, hyperbola)*.
So, what is the physical significance of the determinant $\Delta$ in the general equation of a conic? By physical significance, I mean what happens in the system of intersecting plane and a double right circular cone? I am guessing that the value $\Delta$ represents some kind of distance of the slicing plane from the vertex of the double cone. But it would be great if you could confirm that. Further, is the value of $\Delta$ always positive and zero, or it takes negative value too?
Kindly explain your answer in a simple way, so that a high school student could understand. Thank you in advance.
*Related : Quadratic Curve