Unit 5: Conic Sections
Table of Contents
Introduction to Conic Sections
A conic section (or conic) is a curve obtained by intersecting a plane with a double-napped cone. The shape of the curve depends on the angle of the plane.
A conic can also be defined in 2D as the locus of a point P that moves so that the ratio of its distance from a fixed point (the focus) to its perpendicular distance from a fixed line (the directrix) is a constant value e, called the eccentricity.
Distance from Focus / Distance from Directrix = e - If
e = 1, the conic is a Parabola. - If
e < 1, the conic is an Ellipse. - If
e > 1, the conic is a Hyperbola. - If
e = 0, it is a Circle (a special case of an ellipse).
The Parabola
Definition (e=1)
A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix).
[Image of a parabola, showing the focus, directrix, vertex, and axis of symmetry]Standard Forms and Properties
The "various forms" refer to the four standard orientations of a parabola with its vertex at the origin (0, 0).
| Property | y² = 4ax(Opens Right) | y² = -4ax(Opens Left) | x² = 4ay(Opens Up) | x² = -4ay(Opens Down) |
|---|---|---|---|---|
| Vertex | (0, 0) | (0, 0) | (0, 0) | (0, 0) |
| Focus | (a, 0) | (-a, 0) | (0, a) | (0, -a) |
| Directrix | x = -a | x = a | y = -a | y = a |
| Axis | y = 0 (x-axis) | y = 0 (x-axis) | x = 0 (y-axis) | x = 0 (y-axis) |
| Latus Rectum | 4a | 4a | 4a | 4a |