Consider again the special case that the center of a circle is the
origin and its radius is
.
Then its equation can be rewritten as

This describes an ** ellipse** with its center at the origin,
a ** horizontal axis** of length , and a ** vertical axis** of length
. If we obtain again the circle of radius
(around the origin). The longer of the two axes is called the **
major axis** of the ellipse, and the shorter its ** minor axis**.
Textbooks often assume that (and switch and if the
vertical axis is the major axis).

** Figure 1. Circles and Ellipses**

Figure 1 shows the unit circle in red, the ellipse with
in yellow, and the ellipse with
in green.

An ellipse can be translated so that its center is at
) which gives rise to the equation

which is sometimes called the ** standard equation of an ellipse.**
Note how all the relevant information about the ellipse, i.e., its
center and the lengths and orientations of its major and minor axis,
can be read directly from this equation.

** Figure 2. Two Hyperbolas and their Asymptotes**

Consider again the equation of the unit circle. If we replace
addition by subtraction we obtain the equations

or
Each of these these equations describes
a ** hyperbola**. A hyperbola has two branches. In Figure 2, the
yellow and blue curves form the graph of the purple and
turquoise curves form the graph of . Also shown are the lines
(red) and (green) which are the ** asymptotes** of the
hyperbolas. The graphs of the hyperbolas approach these lines
arbitrarily closely without ever reaching them.
Like other graphs, hyperbolas can be scaled and translated.

### Conic Sections

The discussion on this page is a very brief introduction to a much
larger and richer subject: the study of ** conic sections**. A conic
section is the intersection of a plane and a cone. Such curves can be
expressed as the graphs of the general quadratic equation

where are constants.
Conic sections include ellipses, parabolas, hyperbolas, pairs of
intersecting lines, and single points. Their detailed study is
intricate, fruitful, and fascinating, but beyond the scope of this
class.