Given a general Quadratic Curve
|
(1) |
the quantity is known as the discriminant, where
|
(2) |
and is invariant under Rotation. Using the Coefficients from Quadratic
Equations for a rotation by an angle ,
Now let
and use
to rewrite the primed variables
From (11) and (13), it follows that
|
(14) |
Combining with (12) yields, for an arbitrary
which is therefore invariant under rotation. This invariant therefore provides a useful shortcut to determining the
shape represented by a Quadratic Curve. Choosing to make (see Quadratic Equation), the
curve takes on the form
|
(16) |
Completing the Square and defining new variables gives
|
(17) |
Without loss of generality, take the sign of to be positive. The discriminant is
|
(18) |
Now, if , then and both have the same sign, and the equation has the general form of an
Ellipse (if and are positive). If , then and have opposite signs, and the equation has
the general form of a Hyperbola. If , then either or is zero, and the equation has the
general form of a Parabola (if the Nonzero or is positive). Since the discriminant is invariant, these
conclusions will also hold for an arbitrary choice of , so they also hold when is replaced by the
original . The general result is
- 1. If , the equation represents an Ellipse, a Circle (degenerate Ellipse),
a Point (degenerate Circle), or has no graph.
- 2. If , the equation represents a Hyperbola or pair of intersecting lines (degenerate
Hyperbola).
- 3. If , the equation represents a Parabola, a Line (degenerate Parabola),
a pair of Parallel lines (degenerate Parabola), or has no graph.
© 1996-9 Eric W. Weisstein
1999-05-24