Parabola definition parboiling my frontal lobe!!

ngm01

1. The problem statement, all variables and given/known data

The general equation of second degree is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. the values of the coefficients in the general eqn. for a parabola shown above could be:

A B C D E F
+ve 0 0 0 +ve 0
-ve 0 0 0 +ve 0
0 0 +ve +ve 0 0
0 0 +ve -ve 0 0 *** This is the combination that works???
0 -ve +ve 0 0 0



2. Relevant equations

Conic section equation where from the general equation:
B^2 - AC = 0 (Parabola)
B^2-AC < 0 (Ellipse),
B^2-AC > 0 (Hyperbola),
A=C and B=0 (Circle)
A=B=C=0 straight line



3. The attempt at a solution

I started by using the conic section coefficient guide for a parabola and substituted the possible values of the coefficients given in the table above into the conic section equation to try to eliminate a combination that would be a ellipse or hyperbola and the infer by the process of elimination that...." Aha these coefficients can define a hyperbola so it can't work!!...." but they all satisfy pretty much any of the guides --so that went down in flames, I really don't see what makes that combination of coefficients special for a parabola. So if you do please help me understand.

Regards