Conic Sections Basics

Conic Sections

Definition

• A conic section is a curve obtained by intersecting a cone with a plane
• The resulting curve depends on the angle of intersection between the plane and the cone

Types of Conic Sections

• Circle: formed when the plane is perpendicular to the cone's axis
  • Equation: x^2 + y^2 = r^2 (standard form)
  • Characteristics: constant radius, no vertex, no focal points
• Ellipse: formed when the plane is inclined at an angle less than 90° to the cone's axis
  • Equation: (x^2/a^2) + (y^2/b^2) = 1 (standard form)
  • Characteristics: two foci, major and minor axes, eccentricity (e)
• Parabola: formed when the plane is parallel to the cone's generator
  • Equation: y = ax^2 + bx + c (standard form)
  • Characteristics: one focus, vertex, axis of symmetry
• Hyperbola: formed when the plane is inclined at an angle greater than 90° to the cone's axis
  • Equation: (x^2/a^2) - (y^2/b^2) = 1 (standard form)
  • Characteristics: two foci, transverse and conjugate axes, eccentricity (e)

Key Concepts

• Focus: a point on the conic section that is equidistant from every point on the curve
• Vertex: the lowest or highest point on the conic section (depending on orientation)
• Axis of symmetry: the line that passes through the vertex and is perpendicular to the directrix
• Directrix: a line that is perpendicular to the axis of symmetry and is used to define the conic section
• Eccentricity (e): a measure of how "stretched" the conic section is (e = 0 for a circle, e > 0 for an ellipse, e = 1 for a parabola, e > 1 for a hyperbola)