Overview
This lecture explains conic sections, their types, and how they are formed when a plane intersects a double right circular cone at various angles.
Structure of a Double Right Circular Cone
- A double right circular cone is formed when a line rotates about a fixed point (vertex) and intersects a vertical line.
- It consists of two cones joined at the vertex, with the fixed line as the axis and the rotating line as the generator.
- The base is circular, and its center connects perpendicularly to the vertex via the axis.
- The perimeter of the base is the directrix, and the curved surface is called the nappe.
- There are two nappes: upper (above the vertex) and lower (below the vertex).
- The angle between the generator and the axis is called the vertex angle.
Formation of Conic Sections
- Conic sections are curves formed when a plane intersects a double right circular cone at different angles.
- The type of conic depends on the angle between the intersecting plane and the cone’s axis.
Types of Conic Sections
- Ellipse: Formed when the plane angle with the axis is greater than the vertex angle; a circle is a special case when the plane is perpendicular to the axis.
- Parabola: Formed when the plane angle with the axis is exactly equal to the vertex angle.
- Hyperbola: Formed when the plane angle with the axis is less than the vertex angle, intersecting both nappes to create two disjoint curves.
Degenerate Conic Sections
- If the plane intersects the cone at its vertex, the ellipse reduces to a point, the parabola to a line, and the hyperbola to two intersecting lines.
- These resulting figures are called degenerate conics.
Key Terms & Definitions
- Double Right Circular Cone — two identical cones joined at a vertex with a common axis.
- Vertex — the fixed point where the cones meet.
- Axis — the fixed line through the vertex.
- Generator — the rotating line that forms the cone.
- Vertex Angle — angle between the generator and the axis.
- Directrix — the perimeter of the cone's base.
- Nappe — the curved surface of a cone; upper nappe is above, lower nappe is below the vertex.
- Conic Section — a curve formed by the intersection of a plane and a cone (ellipse, parabola, hyperbola).
- Degenerate Conic — result when a plane passes through the vertex (point, line, or intersecting lines).
Action Items / Next Steps
- Review the definitions and distinguishing features of ellipse, parabola, and hyperbola.
- Practice sketching conic sections formed by various plane angles.