Conic Sections/General Conic Sections

The general form of any conic section is $Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$. In most cases, B=0 because any other value would cause the conic to be rotated and makes analysis much more difficult. For the rest of this page we will assume that B=0. The general equation can be algebraically manipulated to become the standard form of the specific conic it describes. However, it would first be useful to know what that conic is. That can be determined simply by comparing A and C. If either A or C, but not both, equals 0, the equation describes a parabola. If A=C, it describes a circle. If A and C are both positive or both negative, but not equal, it is an ellipse. If A and C have opposite signs, it is a hyperbola. If A=C=0, it's obviously just a line.