The Conics -- Translation and Rotation

I. General Form of the 2nd degree equation in 2 variables:
        Ax2 + Bxy + Cy2 + Dx + Ey + F = 0

    A. Test the discriminant B2 - 4AC
   
If B2 - 4AC < 0 Then Ellipse, Circle, Point, or Nothing
If B2 - 4AC > 0 Then Hyperbola or Two Intersecting Lines
If B2 - 4AC = 0 Then Parabola, Two Parallel Lines, One Line, or Nothing

    B. If B = 0, the equation becomes Ax2 + Cy2 + Dx + Ey + F = 0
If A and C are both positive or both negative Then Ellipse, Circle (if A = C), Point, or Nothing
If A and C are of opposite signs Then Hyperbola or Two Intersecting Lines
If A = 0 or B = 0 Then Parabola, Two Parallel Lines, One Line, or Nothing

II. Translations from xy-axes to x'y'-axes

          x' = x - h
y' = y - k
or x = x' + h
y = y' + k

III. Rotations from xy-axes to x'y'-axes