Nondegenerate conic sections

Definition

Nondegenerate conic sections are the curves obtained by intersecting a plane with a double-napped cone, which do not degenerate into simpler forms. These include ellipses, parabolas, and hyperbolas.

5 Must Know Facts For Your Next Test

1. Nondegenerate conic sections can be classified as ellipses, parabolas, or hyperbolas based on the angle of intersection between the plane and the cone.
2. The general second-degree equation for nondegenerate conic sections is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
3. Rotation of axes can simplify the general equation of a conic section by eliminating the $Bxy$ term.
4. The discriminant $B^2 - 4AC$ determines the type of conic: ellipse ($<0$), parabola ($=0$), or hyperbola ($>0$).
5. Conic sections have important geometric properties such as foci, directrices, and eccentricity.

Review Questions

• What are the three types of nondegenerate conic sections?
• How does rotation of axes affect the equation of a nondegenerate conic section?
• What does the discriminant $B^2 - 4AC$ indicate about a conic section?

Related terms

Ellipse: A type of nondegenerate conic section formed when a plane intersects a cone at an angle less than that made by the side of the cone with its base.

Parabola: A type of nondegenerate conic section formed when a plane is parallel to one generating line of the cone.

Hyperbola: A type of nondegenerate conic section formed when a plane intersects both nappes (opposite cones) at an angle steeper than that made by the side of the cone with its base.