and [Back] [VVHS Math - Mr. Davis]
Definition: An ellipse is a locus (collection) of points such that: For any point P(x,y) on the ellipse, the sum of the distances to two given points F1 and F2 (the foci) is a constant.
PF1 + PF2 = 2a
The ellipse is defined like the circle, as a locus of points.
However, an ellipse has two centers or foci (the
plural of focus), and instead of a radius as the given distance, an ellipse
has a focal constant (also known as the sum
of the focal radii) as its given constant distance.
The ellipse below has foci at F2
and
F1
. The focal constant is 10.
The sum of the two distances to the two foci from each point P is always 10. One distance grows larger as the other gets shorter, keeping the sum at a constant.
While a circle has a constant diameter, an ellipse has two lengths which determine its size, the major and minor axes. The vertices of an ellipse are the endpoints of the major axis. The foci of the ellipse always lie on the major axis.
For an ellipse with center (0,0), the standard form equation is:
Substituting 0 for x and y gives x-intercepts
and 
, and y-intercepts 
and 
.
The foci are located at 
and
using the formula 
. Memorize it!
The ellipse shown has equation 
The vertices (x-intercepts) are V2 
and V1 
.
The endpoints of the minor axis (y-intercepts)
are 
and 
.
Using the formula 
, the distance from the center to the foci is 
, or 
. Therefore the foci are F2 
and F1 
.
Major Axis = 10
Minor Axis = 6
Center (0,0)
Major Axis Horizontal
Vertices: (a, 0) (-a, 0)
Foci: (c, 0) (-c, 0)
Equation: 
Major Axis: 2a
Minor Axis: 2b
Distance between foci: 2c
Distance from foci to center: 
Major Axis Vertical
Vertices: (0, a) (0, -a)
Foci: (0, c) (0, -c)
Equation: 
Major Axis: 2a
Minor Axis: 2b
Distance between foci: 2c
Distance from foci to center: 
Focal Constant = 2a
The focal constant is equal to the major
axis.
By definition, the sum of the distances
to the foci (the focal constant) is the same for every point on
the ellipse. Since the point (a,0) is on the ellipse, the sum of the distances
from (a,0) to the foci (c,0) and (-c,0) equals the focal constant. This
distance is:
(a+c)+(a-c) = focal constant
2a = focal constant
The relationship between a, b, and c is given by the following formula:
Once again, since (0,b)is on the ellipse, the sum of the
distances to the foci equals the focal constant. Since the distance from
(0,b) to each focus is equal, the distance from (0,b) to each focus must
equal a.
This creates a right triangle with legs of length b and
c, and hypotenuse of length a, giving the
relation
b2 + c2 = a2, or c2 = a2 - b2 ,
or 
The equation of a translated ellipse is
similar to the center-radius form equation of a circle.
An ellipse with center 
has equation
The parts of the ellipse are found by adding the center coordinates onto the normal (untranslated) points.
Center (h,k)
Major Axis Horizontal
Vertices: (a+h, k) (-a+h, k)
Foci: (c+h, k) (-c+h, k)
Major Axis: 2a
Minor Axis: 2b
Distance between foci: 2c
Distance from foci to center: 
The graph below shows an example of a translated ellipse. The equations of the two ellipses are
and 
Parent Graph: Center 
Vertices 
Foci 
Translated Graph: Center 
Vertices
or 
, 
Foci 
or 
,