Quadratic Curve Discriminant

Given a general quadratic curve

the quantity X is known as the discriminant, where

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

Now let

and use

to rewrite the primed variables

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

Combining with (17) yields, for an arbitrary theta

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

Completing the square and defining new variables gives

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

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

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Cite this as:

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

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