Degenerate conic sections

Definition

Degenerate conic sections are special cases of conic sections that result when the plane intersects the cone in a manner that produces simpler geometric figures like points, lines, or pairs of lines. They occur when the discriminant of a conic section's equation is zero.

5 Must Know Facts For Your Next Test

1. A degenerate conic can be represented by equations such as $Ax^2 + Bxy + Cy^2 = 0$.
2. Degenerate conics include points, lines, and intersecting lines.
3. They occur when the determinant of the quadratic form matrix is zero, leading to a rank-deficient matrix.
4. Rotation of axes may transform a non-degenerate conic into a degenerate one under specific conditions.
5. Analyzing the discriminant $B^2 - 4AC$ helps determine whether a conic section is degenerate.

Review Questions

• What types of geometric figures result from degenerate conic sections?
• How does the discriminant $B^2 - 4AC$ relate to identifying degenerate conics?
• Explain how rotation of axes can affect whether a conic section becomes degenerate.

Related terms

Conic Section: Curves obtained by intersecting a plane with a double-napped cone, including circles, ellipses, parabolas, and hyperbolas.

Discriminant: $B^2 - 4AC$; used to determine the nature (ellipse, parabola, hyperbola) or degeneracy of a conic section.

Rotation of Axes: A transformation technique used to eliminate the $xy$-term in the equation of a conic section.