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I’m studying conic sections, and this question had me stumped.

What restriction should be put on the constants so that the general equation for any conic section, $Ax^2+ Bxy+ Cy^2+ Dx + Ey +F =0$ will always represent a conic?

I’ve scoured the Internet but can’t find any webpage which has the answer to this question.

I thought of the restrictions $A≠0$ and $C≠0$, but I came across a degenerate conic (a line) in which $A=0$ and $C=0$. In this case, the restrictions $A≠0$ and $C≠0$ would no longer be suitable, unless a degenerate conic is not considered as a conic.

Or, if rotated conics are not considered as conics, then a suitable restriction would be $B=0$.

Are my assumptions correct?

gc3941d
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    Does this help: https://www.math.utah.edu/~wortman/1060text-coc.pdf – ancient mathematician Dec 19 '20 at 15:54
  • Yes, it does. It says that $A,B$ and $C$ don’t all equal zero. How can I put this sentence in mathematical notation? @ancientmathematician – gc3941d Dec 19 '20 at 16:05
  • Personally that seems to me a fine sentence in mathematical english. You could write $(A,B,C)\not=(0,0,0)$ but I wouldn't. – ancient mathematician Dec 19 '20 at 16:08
  • I see, thanks for your help! – gc3941d Dec 19 '20 at 16:10
  • Note that if $A=B=C=0,$ the equation describes either a line, the empty set, or the whole plane. The first two shapes are recognized as degenerate conics by https://www.math.utah.edu/~wortman/1060text-coc.pdf. So this condition on $A,B,C$ distinguishes a quadratic equation from a lower-degree equation, but says very little about what the equation represents. It rules out the case where all coefficients are zero, which is solved by all $(x,y)$; every other shape represented by an equation of one or zero degrees is also represented by a quadratic. – David K Dec 19 '20 at 17:29
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    If "conic" in the question actually means "non-degenerate conic", then "$A$, $B$, and $C$ not all zero" is a necessary condition but not a sufficient condition. I describe necessary and sufficient conditions at the end of this answer. – David K Dec 19 '20 at 17:37
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    If degenerate conics such as one or two lines are accepted, then we need a definition of degenerate conic. If a point, the empty set, and the entire plane are also considered degenerate conics then there are no restrictions on the coefficients in the equation. – David K Dec 19 '20 at 17:49
  • @DavidK Thank you for the references. They are indeed helpful. – gc3941d Dec 20 '20 at 04:53
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    @gc3941d see "Classification" in this wikipedia article https://en.wikipedia.org/wiki/Matrix_representation_of_conic_sections – user376343 Dec 20 '20 at 18:44

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