A question on conic sections?

Question

Hey guys, there is the general equation for a conic section:

Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

For any arbitrary value of A, B, C, D, E & F how would you determine if the equation represents:

1. A circle
2. A pair of lines
3. A parabola
4. An ellipse
5. A hyperbola

Sorry if it is too long, but maybe there is some general set of conditions to be checked for, in this case.

Answer 1

* if B2 − 4AC < 0, the equation represents an ellipse (unless the conic is degenerate, for example x2 + y2 + 10 = 0);
* if A = C and B = 0, the equation represents a circle;
* if B2 − 4AC = 0, the equation represents a parabola;
* if B2 − 4AC > 0, the equation represents a hyperbola;
*if we also have A + B = 0, the equation represents a rectangular hyperbola (two lines).

Answer 2

circle: A=C; parabola A or C=0; ellipse AB>0; hyperbola AB<0; pair of lines A=C=0