TOPICS
Search

Quadratic Curve Discriminant


Given a general quadratic curve

 Ax^2+Bxy+Cy^2+Dx+Ey+F=0,
(1)

the quantity X is known as the discriminant, where

 X=B^2-4AC,
(2)

and is invariant under rotation. Using the coefficients from quadratic equations for a rotation by an angle theta,

A^'=1/2A[1+cos(2theta)]+1/2Bsin(2theta)+1/2C[1-cos(2theta)]
(3)
=(A+C)/2+B/2sin(2theta)+(A-C)/2cos(2theta)
(4)
B^'=Gcos(2theta+delta-pi/2)=Gsin(2theta+delta)
(5)
C^'=1/2A[1-cos(2theta)]-1/2Bsin(2theta)+1/2C[1+cos(2theta)]
(6)
(7)
=(A+C)/2-B/2sin(2theta)+(C-A)/2cos(2theta).
(8)

Now let

G=sqrt(B^2+(A-C)^2)
(9)
delta=tan^(-1)(B/(C-A))
(10)
delta_2=tan^(-1)((A-C)/B)
(11)
=-cot^(-1)(B/(C-A)),
(12)

and use

cot^(-1)(x)=1/2pi-tan^(-1)(x)
(13)
delta_2=delta-1/2pi
(14)

to rewrite the primed variables

A^'=(A+C)/2+1/2Gcos(2theta+delta)
(15)
B^'=Bcos(2theta)+(C-A)sin(2theta)
(16)
=Gcos(2theta+delta_2)
(17)
C^'=(A+C)/2-1/2Gcos(2theta+delta).
(18)

From (16) and (18), it follows that

 4A^'C^'=(A+C)^2-G^2cos(2theta+delta).
(19)

Combining with (17) yields, for an arbitrary theta

X=B^('2)-4A^'C^'
(20)
=G^2sin^2(2theta+delta)+G^2cos^2(2theta+delta)-(A+C)^2
(21)
=G^2-(A+C)^2=B^2+(A-C)^2-(A+C)^2
(22)
=B^2-4AC,
(23)

which is therefore invariant under rotation. This invariant therefore provides a useful shortcut to determining the shape represented by a quadratic curve. Choosing theta to make B^'=0 (see quadratic equation), the curve takes on the form

 A^'x^2+C^'y^2+D^'x+E^'y+F=0.
(24)

Completing the square and defining new variables gives

 A^'x^('2)+C^'y^('2)=H.
(25)

Without loss of generality, take the sign of H to be positive. The discriminant is

 X=B^('2)-4A^'C^'=-4A^'C^'.
(26)

Now, if -4A^'C^'<0, then A^' and C^' both have the same sign, and the equation has the general form of an ellipse (if A^' and B^' are positive). If -4A^'C^'>0, then A^' and C^' have opposite signs, and the equation has the general form of a hyperbola. If -4A^'C^'=0, then either A^' or C^' is zero, and the equation has the general form of a parabola (if the nonzero A^' or C^' is positive). Since the discriminant is invariant, these conclusions will also hold for an arbitrary choice of theta, so they also hold when -4A^'C^' is replaced by the original B^2-4AC. The general result is

1. If B^2-4AC<0, the equation represents an ellipse, a circle (degenerate ellipse), a point (degenerate circle), or has no graph.

2. If B^2-4AC>0, the equation represents a hyperbola or pair of intersecting lines (degenerate hyperbola).

3. If B^2-4AC=0, the equation represents a parabola, a line (degenerate parabola), a pair of parallel lines (degenerate parabola), or has no graph.


See also

Quadratic

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Quadratic Curve Discriminant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/QuadraticCurveDiscriminant.html

Subject classifications