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
The value of $\Delta$ has no significance. You can scale all the terms in the equation of the conic by a constant factor $\lambda$ and they still describe the same conic. Doing so scales $\Delta$ by $\lambda^3$. So unless you have further constraints (and I can't think of natural constraints you might have), the scale of the coefficients and the value of the determinant are pretty much arbitrary.
In this sense the equation is homogeneous in the conic coefficients (but not in the coordinates $(x,y)$ the way you wrote it). For homogenous coordinates the overall magnitude is arbitrary, it's only the ratios between the individual coordinates that describe the object.