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.
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.