Overview
This lecture introduces the different types of conic sections, which are curves formed by the intersection of a plane and a right circular cone.
Parts of a Right Circular Cone
- The generator is the slanting height of the cone.
- The vertex is the tip point where the generators meet.
- The axis is the vertical line passing through the vertex.
Conic Sections: Types and How They Are Formed
- Conic sections are curves formed by the intersection of a plane and a right circular cone.
- If the cutting plane is parallel to the generator, the resulting curve is a parabola.
- If the cutting plane is perpendicular to the axis, the resulting curve is a circle.
- If the cutting plane is slanted but not parallel to the generator, the resulting curve is an ellipse.
- If the plane intersects both nappes (sides) of the cone, the resulting curve is a hyperbola.
Degenerate Conics
- Degenerate conics are special cases where the intersection results in a point or a line.
- When the plane passes through the cone's vertex, the intersection may be a single point or a line.
- In the hyperbola case, cutting through the vertex with the plane along the axis forms two intersecting lines.
- Degenerate conics are typically not included among the four main conic sections.
The Four Main Conic Sections Studied
- Circle
- Ellipse
- Parabola
- Hyperbola
Key Terms & Definitions
- Conic Section — A curve formed by the intersection of a plane and a right circular cone.
- Generator — The slant height (side) of the cone.
- Axis — The vertical line through the cone’s vertex.
- Vertex — The tip point where the cone’s sides meet.
- Degenerate Conic — A special intersection resulting in a point or line instead of a standard curve.
Action Items / Next Steps
- Review the properties and shapes of each conic section.
- Prepare questions for further clarification in the next class.