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Given the general equation $Ax^2 + Cy^2 + Dx + Ey + F = 0$, what constraints on the set $\{A,C,D,E,F\}$ will apply if the equation represents a (a) parabola? (b) ellipse? (c) hyperbola?

Firstly, I understand in the case of a parabola that $A$ and $C$ cannot both be non-zero at the same time. I also know in the case of a hyperbola that $A$ and $C$ will have opposite signs; and also that for an ellipse, $A$ and $C$ will be different numbers but with the same sign (a circle will be the same number).

I seek clarification as to the reasons for the ellipse and the hyperbola cases.

Also, would I need to go into more depth for each one? I feel I may have to. For example, in the case of a parabola, $b^2 - 4ac >0$, etc.

Thanks very much.

user2768428
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    Hint: Begin by writing out the usual form of the equations defining these conic sections, then rearrange them into the above form. – angryavian Aug 04 '14 at 14:22
  • The equation is not so general since the term in $xy$ is missing - is that done on purpose? In this way you are considering only the conic sections with axis parallel to the coordinate ones. – Jack D'Aurizio Aug 04 '14 at 14:22
  • The term in xy is missing on purpose. Thanks for the suggestions Jack and angry – user2768428 Aug 04 '14 at 14:36

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If $AC=0$ while $A^2+C^2\neq 0$ you are in the parabolic case.

If $AC<0$ you are in the hyperbolic case.

If $AC>0$ and the equivalent quadratic form $$ |A|(x-x_0)^2 + |C|(y-y_0)^2 = G$$ has a positive $G$, you are in the elliptic case (circular case if $|A|=|C|$).

If $G=0$ the conic is made of a point only (the center $(x_0,y_0)$), if $G<0$ the conic is empty.

Jack D'Aurizio
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  • So once I've established each of these, do I need to go deeper for each one as I suggested earlier, or does this suffice? Thanks – user2768428 Aug 04 '14 at 14:40
  • If your goal is to find the conic type and the sign of $AC$ tells you almost everything, how would you "go deeper"? On what purpose? – Jack D'Aurizio Aug 04 '14 at 14:45
  • Hmm OK I see your point. Could I possibly ask you to justify why for each of the elliptical and hyperbolic cases those properties you mention must be satisfied? – user2768428 Aug 04 '14 at 14:51
  • If $AC<0$, there are points on the conic with arbitrarily large values of $x$ and $y$. Hence the conic is unbounded, and since it is symmetric with respect to its center $(x,y)\in X \Rightarrow (2x_0-x,2y_0-y)\in C$ it is an hyperbola. – Jack D'Aurizio Aug 04 '14 at 14:55
  • If $AC=0$ the conic is still unbounded but it does not have a center, hence it is a parabola. If $AC>0$ and $G>0$, the conic is bounded and non-empty, hence an ellipse. – Jack D'Aurizio Aug 04 '14 at 14:56
  • That's fantastic, thanks. Just the one final question: For any of these conics, do the parameters D, E and F have any constraints? Can they be anything? – user2768428 Aug 05 '14 at 02:46