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 (17) and (19), it follows that
 |
(20)
|
Combining with (18) 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
 |
(27)
|
Completing the square and defining new variables gives
 |
(28)
|
Without loss of generality, take the sign of
to be positive. The discriminant is
 |
(29)
|
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.