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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

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    $\begingroup$ (1) It is not true that if $\Delta=0$, then the equation represents a pair of straight lines. Consider $x^2+y^2=0$ which represents one point $(0,0)$. (2) It is not true that if $\Delta\not=0$, then the equation represents a non-degenerate conic. Consider $x^2+y^2=-1$ which is nothing. (3) $\Delta$ can be negative. Consider $ax^2+by^2+c=0$ with $abc\lt 0$. $\endgroup$ – mathlove Nov 7 '19 at 11:59
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    $\begingroup$ A degenerate conic need not be a pair of straight lines in all cases. It might be a double line or, depending on the field on which you define this conic, even a single point or the empty set. $\endgroup$ – Caligula Nov 7 '19 at 12:00
  • $\begingroup$ @mathlove, Thank you for your comment :) (1) Understood. It is the case the slicing plane is parallel to the base of the cones and passes through the vertex. (2) Thanks. I didn't realise it. An imaginary circle with real centre. (3) Thank you. - Now could you please tell whether the value of $\Delta$ have any physical significance, excluding these cases? When typing this question, I just imagined the cases when the slicing plane cuts the double-cone system at the vertex. Thanks for mentioning the exceptional cases :) $\endgroup$ – Guru Vishnu Nov 7 '19 at 12:09
  • $\begingroup$ @Caligula, Thank you for your comment :) - I don't really know the rigorous definition of degenerate conic. I thought even coinciding lines, and intersecting lines with real point of intersection are called pair of straight lines. But, I don't understand how degenerate case give an empty set, as I think if we are given a plane and a double-cone, they must intersect at atleast one point. $\endgroup$ – Guru Vishnu Nov 7 '19 at 12:12
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    $\begingroup$ I've never heard of the physical significance of $\Delta$. Are you aware of the fact that we cannot distinguish conics only by $\Delta$? For example, each of $x^2+2y^2-1=0$ (ellipse), $x^2-2y^2+1=0$ (hyperbola), $8x^2+y=0$ (parabola) has $\Delta=-2$. Basically, we use $J$ (see the table of the page) to distinguish conics. $\endgroup$ – mathlove Nov 8 '19 at 5:58
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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.

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  • $\begingroup$ Thank you for your answer. Do you think, $\Delta$ must be something related to the distance of the vertex of the double-cone from the slicing plane? I guessed this, as the distance equals zero when the plane passes through this point but non-zero in other cases. As @mathlove mentioned in the comments $\Delta$ can be negative, so I think it must be proportional to an odd power (not even) of distance or any other related quantity. $\endgroup$ – Guru Vishnu Nov 8 '19 at 6:27

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