The Conics — Translation
and Rotation
by David Pleacher

I. General Form of the 2^nd 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

Send any comments or questions to: David Pleacher