Given a general Quadratic Curve
![\begin{displaymath}
Ax^2+Bxy+Cy^2+Dx+Ey+F = 0,
\end{displaymath}](d2_815.gif) |
(1) |
the quantity
is known as the discriminant, where
![\begin{displaymath}
X \equiv B^2-4AC,
\end{displaymath}](d2_817.gif) |
(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
![\begin{displaymath}
4A'C' = (A+C)^2-G^2\cos(2\theta+\delta).
\end{displaymath}](d2_839.gif) |
(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
![\begin{displaymath}
A'x^2+C'y^2+D'x+E'y+F = 0.
\end{displaymath}](d2_846.gif) |
(16) |
Completing the Square and defining new variables gives
![\begin{displaymath}
A'x'^2+C'y'^2=H.
\end{displaymath}](d2_847.gif) |
(17) |
Without loss of generality, take the sign of
to be positive. The discriminant is
![\begin{displaymath}
X = B'^2-4A'C' = -4A'C'.
\end{displaymath}](d2_849.gif) |
(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