http://www.ck12.org Chapter 9. Conics
The general equation for an ellipse is:
(x−h)^2
a^2 +
(y−k)^2
b^2 =^1
In this case the major axis is horizontal becausea, thex-radius, is larger. If they-radius were larger, thenaandb
would reverse. In other words, the coefficientaalways comes from the length of the semi major axis (the longer
axis) and the coefficientbalways comes from the length of the semi minor axis (the shorter axis).
In order to find the locations of the two foci, you will need to find the focal radius represented ascusing the following
relationship:
a^2 −b^2 =c^2
Once you have the focal radius, measure from the center along the major axis to locate the foci. The general shape
of an ellipse is measured using eccentricity. Eccentricity is a measure of how oval or how circular the shape is.
Ellipses can have an eccentricity between 0 and 1 where a number close to 0 is extremely circular and a number
close to 1 is less circular. Eccentricity is calculated by:
e=ac
Ellipses also have two directrix lines that correspond to each focus but on the outside of the ellipse. The distance
from the center of the ellipse to each directrix line isac^2.
Example A
Find the vertices (endpoints of the major axis), foci and eccentricity of the following ellipse.
25 x^2 + 16 y^2 =^1
Solution:The center of this ellipse is at (0, 0). The semi major axis isa=5 and travels horizontally. This means
that the vertices are at (5, 0) and (-5, 0). The semi-minor axis isb=4 and travels vertically.
25 − 16 =c^2
3 =c