Quadratic Curve Discriminant

Given a general quadratic curve

the quantity is known as the discriminant, where

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 (15) and (17), it follows that

Combining with (16) 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

Completing the square and defining new variables gives

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

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.