Mathematics: Geometry - Conic Sections

Conic Sections
Conic sections are a family of shapes that can be thought of as cross sections of a pair of infinite right cones stacked vertex to vertex. This is easiest to see with a visual representation:

A three-dimensional look at representative conic sections. (Note that a hyperbola intersects both cones.)
A side-on look at representative conic sections. (Note that the parabola is parallel to the slant of the cones.)





In short, a circle is a horizontal cross section, a parabola is a cross section parallel to the slant of the cone, an ellipse is a cross section at an angle less than the slant of the cone, and a hyperbola is a cross section at an angle greater than the slant of the cone.

Ellipse

An ellipse is the set of all points in a plane whose total distance from two fixed points called the foci (singular: focus) is constant, and whose center is the midpoint between the foci.

The standard equation of an ellipse that is taller than it is wide is
are coefficients. The center is the point (<i>h</i>, <i>k</i>) and the foci are the points 
and
, where

and
.

The major axis has length 2a, and the minor axis has length 2b.

Eccentricity (e) is a measure of how elongated an ellipse is, and is the ratio of the distance between the foci to the length of the major axis. Eccentricity will have a value between 0 and 1. The closer to 1 the eccentricity is, the closer the ellipse is to being a circle. The formula for eccentricity is

.

Parabola
A parabola is the set of all points in a plane that are equidistant from a fixed line, called the directrix, and a fixed point not on the line, called the focus. The axis is the line perpendicular to the directrix that passes through the focus.
For parabolas that open up or down, the standard equation is
and <i>k</i> are coefficients. If <i>c</i> is positive, the parabola opens up. If <i>c</i> is negative, the parabola opens down. The vertex is the point (<i>h</i>, <i>k</i>). The directrix is the line having the equation <br><img data-cke-saved-src=" />, and the focus is the point
.
For parabolas that open left or right, the standard equation is
and <i>h</i> are coefficients. If <i>c</i> is positive, the parabola opens to the right. If <i>c</i> is negative, the parabola opens to the left.<br> The vertex is the point (<i>h</i>, <i>k</i>). The directrix is the line having the equation <br><img data-cke-saved-src=" />, and the focus is the point
.

Hyperbola
A hyperbola is the set of all points in a plane, whose distance from two fixed points, called foci, has a constant difference.

The standard equation of a horizontal hyperbola is
,

h, and k are real numbers.

The center is the point (h, k), the vertices are the points 
and
,

and the foci are the points that every point on one of the parabolic curves is equidistant from. The foci are found using the formulas

and
, where
.

The asymptotes are two lines the graph of the hyperbola approaches but never reaches, and are given by the equations

and
.
 

The standard equation of a vertical hyperbola is
<i>k</i>, and <i>h</i> are real numbers. The center is the point (<i>h</i>, <i>k</i>), the vertices are the points <br><img data-cke-saved-src=" />
and
, and the foci are the points that every point on one of the hyperbolic curves is equidistant from and are found using the formulas

and
, where
. The asymptotes are two lines the graph of the hyperbola approaches but never reaches, and are given by the equations

and
.