What are Degenerate Conics?
Degenerate conics are special cases of conic sections that occur when the intersecting plane passes through the vertex of the cone in such a way that the resulting figure is simpler and does not form the usual conic section shapes (circles, ellipses, parabolas, or hyperbolas). Instead, they form less complex figures.
Examples of Degenerate Conics
| Degenerate Conics | Description | Real-World Examples |
|---|---|---|
| Point | A single point formed when the plane intersects the vertex of the cone and does not pass through any other part of the cone. | The tip of a sharpened pencil or the origin in a coordinate system. |
| Line | A straight line formed when the plane intersects the cone through its side and passes through the vertex. | The beam of a flashlight viewed edge-on or a taut string. |
| Intersecting Lines | Two lines that intersect at the vertex, formed when the plane passes through the vertex and intersects both nappes of the cone. | Street intersections viewed from above or X-shaped cross-bracing in architecture. |
Degenerate and Non-Degenerate Conics
Conic sections, or simply conics, are shapes created by cutting a cone with a plane. These shapes include circles, ellipses, parabolas, and hyperbolas, each with unique properties and equations. Conics can be broadly classified into two categories: degenerate and non-degenerate conics.
Non-degenerate conics are the typical conic sections most people are familiar with, such as circles, parabolas, ellipses, and hyperbolas. On the other hand, degenerate conics occur when the plane cuts through the cone in a way that results in simpler or more ‘collapsed’ shapes, such as points, lines, and intersecting lines.
Let’s dicusss degenerate and non-degenerate conics in this article in detail.
Table of Content
- What are Conic Sections?
- Classification of Conics
- What are Degenerate Conics?
- Examples of Degenerate Conics
- What are Non-Degenerate Conics?
- Examples of Non-Degenerate Conics
- Differences Between Degenerate and Non-Degenerate Conics
- Summary
- FAQs on Degenerate and Non-Degenerate Conics