Keo's Notes

LEGACY Conic Sections

3 min read

Definition

Conic sections are graphs that are obtained by slicing a cone (with an inverted cone on top) with a plane Conic Sections

A-Level Further Pure 1

Eccentricity

For all points on a conic section, the ratio of the distance from a fixed point (the focus) and fixed straight line (the directrix) is constant. This ratio is called the eccentricity.

  • If , describes an ellipse
  • If , describes a parabola
  • If , describes a hyperbola

Summary of Conics and Properties

PropertyEllipseParabolaHyperbolaRectangular Hyperbola
Standard Form
Parametric Form
Eccentricity

Foci
Directrices
Asymptotesnonenone

Parabolas

Formed by slicing a cone with a plane parallel to its slant Cartesian Equation: Parametric Equation: , for Eccentricity: Focus: Directrix:

Hyperbolas

Formed by slicing a cone with a plane that intersects both nappes of the cone Cartesian Equation: Parametric Equation: , or , Eccentricity: Focus: Directrix:

Other Properties where is the focal length meaning the distance from the centre to a focus

Rectangular Hyperbolas

Formed by slicing a cone with a plane perpendicular to the axis of symmetry Cartesian Equation: Parametric Equation: Eccentricity: Focus: Directrix:

Ellipses

For major axis on the -axis meaning - If the major axis is the y-axis so swap a and b around for everything Formed by slicing a cone with a plane at an angle less than that of the cone’s slant Cartesian Equation: Parametric Equation: , Eccentricity: Focus: Directrix:

Other Properties where is the focal length meaning the distance from the centre to a focus

Graph View