Degenerate conic sections

Definition

Degenerate conic sections are special cases of conic sections that do not form the usual shapes like ellipses, parabolas, or hyperbolas. They occur when the plane intersects the cone at its vertex or in other ways that produce a single point, a line, or intersecting lines.

5 Must Know Facts For Your Next Test

1. Degenerate conic sections include a single point, a line, and intersecting lines.
2. They result from specific conditions in the general quadratic equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$.
3. When $\Delta = B^2 - 4AC = 0$, and the determinant $AD - BC + E^2/4 = 0$, it often indicates a degenerate case.
4. A degenerate ellipse can occur as either a point or an empty set.
5. In analytic geometry, degenerate conics are important for understanding the full scope of solutions to quadratic equations.

Review Questions

• What conditions in the general quadratic equation indicate a degenerate conic section?
• List all possible forms that a degenerate conic section can take.
• Explain how rotation of axes might lead to identifying degenerate conic sections.

Related terms

Conic Sections: Curves obtained by intersecting a right circular cone with a plane; includes circles, ellipses, parabolas, and hyperbolas.

Quadratic Equation: A polynomial equation of degree two in one variable typically written as $ax^2 + bx + c = 0$.

Rotation of Axes: A technique used to eliminate the $xy$-term in the equation of a conic section by rotating the coordinate system through an angle.