Overview
This lecture explains conic sections, their definitions, and how different intersections between a plane and a double right circular cone create ellipses, parabolas, and hyperbolas.
Double Right Circular Cone and Key Elements
- A double right circular cone is formed by rotating a straight line (generator) about a fixed point (vertex) along a vertical axis.
- The fixed line is the axis, and the perimeter of the cone's base is the directrix.
- The cone has two equal parts called naps: upper nap (above the vertex) and lower nap (below the vertex).
- The vertex angle is the angle between the generator and the axis.
Formation of Conic Sections
- Conic sections arise when a plane intersects a double right circular cone at various angles.
- The type of curve depends on the angle between the plane and the cone's axis.
Types of Conic Sections
- Ellipse: Formed when the plane's angle with the axis is greater than the vertex angle; the ellipse is closed.
- Circle: A special ellipse formed when the plane is perpendicular to the axis.
- Parabola: Formed when the angle between the plane and the axis equals the vertex angle; the curve is open.
- Hyperbola: Formed when the plane's angle with the axis is less than the vertex angle, intersecting both naps and resulting in two separate curves.
Degenerate Conics
- If the plane intersects at the vertex, the resulting figures are degenerate conics: a point (ellipse), a line (parabola), or two intersecting lines (hyperbola).
Key Terms & Definitions
- Vertex — Fixed point where the generator rotates and the cones meet.
- Axis — The stationary straight line about which the generator rotates.
- Generator — The line that rotates to form the cone's surface.
- Directrix — The perimeter of the base of the cone.
- Nap — One part of the double cone, either above or below the vertex.
- Vertex Angle — The angle between the generator and the axis.
- Conic Section — A curve formed by the intersection of a plane with a double right circular cone.
Action Items / Next Steps
- Review the definitions and diagrams of ellipses, parabolas, and hyperbolas.
- Practice identifying conic sections based on the intersection angle.
- Complete relevant textbook exercises on conic sections.