The conic equation is .

The discriminant is defined as .

For this instance, it is .

Since a conic that is rotated as no term, the discriminant reduces to .

Thus, only the and terms determine the type of conic.

If , then the conic represents a parabola.

If , then the conic represents an ellipse or a circle.

If , then the conic represents a hyperbola.

For a circle or an ellipse , and need to share the same sign.

For a parabola, either or has to be equal to .

For a hyperbola, and need to have opposite signs.