Conic Sections Lecture Notes

Introduction to Conic Sections

• Definition: Conic sections are curves obtained by intersecting a plane with a cone.
• Types: Circle, Ellipse, Parabola, Hyperbola

Shapes and Basic Definitions

Circle

• Definition: All points equidistant from a center point.
• Equation:
  • Center:
  • Radius:
• Example:
  • Circle centered at origin, radius r.

Ellipse

• Definition: A squished circle, symmetrical along two axes.
• Relation: Circles are special cases of ellipses.
• Example:
  • General shape like an elongated circle.

Parabola

• Definition: U-shaped curve.
• Equation:
  • y = x² or x = y²
• Example: Classic parabolas opening up/down/left/right.

Hyperbola

• Definition: Two open curves resembling parabolas, but with asymptotes.
• Example:
  • Hyperbolas open in opposite directions.

Why Called Conic Sections?

• Reason: They are the intersection of a plane and a cone.
• Relationship: Circles, ellipses, parabolas, and hyperbolas are all related as different types of intersections.

Visualizing Conic Sections

Circle

• Intersection: Plane perpendicular to the axis of the cone.
• Result: A circle.

Ellipse

• Intersection: Plane tilted slightly, not perpendicular to the cone's axis.
• Result: An ellipse.

Parabola

• Intersection: Plane parallel to the side of the cone, intersecting only one side.
• Result: A parabola.
• Relation: Parabola is a form where an ellipse stretches open on one side.

Hyperbola

• Intersection: Plane intersects both parts of the cone.
• Result: Two curves that form a hyperbola.

Summary

• Conic sections are related: All shapes (circle, ellipse, parabola, hyperbola) can be derived from the intersection of a plane with a cone.
• Next Steps: Upcoming discussion on formulas and graph plotting for conic sections.