Conic Sections

Introduction to Conics

A conic is the intersection of a plane and a right circular cone. The four basic types of conics are parabolas, ellipses, circles, and hyperbolas. We've already discussed parabolas and circles in previous sections, but here we'll define them a new way. Study the figures below to see how a conic is geometrically defined.

Figure %: The four basic types of conics

In the conics above, the plane does not pass through the vertex of the cone. When the plane does intersect the vertex of the cone, the resulting conic is called a degenerate conic. Degenerate conics include a point, a line, and two intersecting lines.

The equation of every conic can be written in the following form: Ax ² + Bxy + Cy ² + Dx + Ey + F = 0 . This is the algebraic definition of a conic. Conics can be classified according to the coefficients of this equation.

The determinant of the equation is B ² - 4AC . Assuming a conic is not degenerate, the following conditions hold true: If B ² -4AC > 0 , the conic is a hyperbola. If B ² -4AC < 0 , the conic is a circle, or an ellipse. If B ² - 4AC = 0 , the conic is a parabola.

Another way to classify conics has to do with the product of A and C . Assuming a conic is not degenerate, the following conditions hold true: if AC > 0 , the conic is an ellipse or a circle. If AC < 0 , the conic is a hyperbola. If AC = 0 , and A and C are not both zero, the conic is a parabola. Finally, if A = C , the conic is a circle.

In the following sections we'll study the other forms in which the equations for certain conics can be written, and what each part of the equation means graphically.