Nondegenerate conic sections

Definition

Nondegenerate conic sections are the curves obtained from the intersection of a plane and a double-napped cone, which include ellipses, parabolas, and hyperbolas. They are distinct from degenerate conics such as points, lines, or intersecting lines.

5 Must Know Facts For Your Next Test

1. Nondegenerate conic sections maintain their shape under rotation of axes.
2. The general second-degree equation for nondegenerate conic sections is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
3. When the discriminant $B^2 - 4AC < 0$, the conic is an ellipse; if it equals zero, it is a parabola; if greater than zero, it is a hyperbola.
4. The rotation of axes can eliminate the $xy$ term by transforming coordinates, making it easier to identify the type of conic section.
5. Ellipses have two foci, hyperbolas have two branches and asymptotes, and parabolas have a single focus and directrix.

Review Questions

• What does the discriminant $B^2 - 4AC$ tell you about the type of nondegenerate conic section?
• How does rotating the axes affect the general form of a conic section's equation?
• What are the key differences in properties between ellipses, parabolas, and hyperbolas?

Related terms

Discriminant: In the context of conic sections, it is given by $B^2 - 4AC$ and determines whether the conic is an ellipse (negative), parabola (zero), or hyperbola (positive).

Rotation of Axes: A transformation that rotates coordinate axes to simplify equations, often used to eliminate the $xy$ term in second-degree equations.

Degenerate Conics: Conics that do not form curves but rather reduce to simpler forms like points or intersecting lines.