MA 15400 Lesson 27 Section 11.2
Ellipses
1
An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points
(the foci) in the plane is a positive constant.
The midpoint of the two foci is the center of the ellipse. The center C of the above ellipse is
marked.
There are two axes: The major axis is the longer of the two and contains the foci. Its endpoints
are the vertices of the ellipses. The minor axis is the shorter of the two and its endpoints are
simply known as the endpoints of the minor axis.
There are three important numbers associated with an ellipse: a, b, and c. These numbers
correspond the distances involving the center, the foci, the vertices, and the endpoints of the
minor axis.
Points
and
F F
′
are the foci
(plural of focus). The sum of
the distances from any point of
an ellipse (such as
P or H) to
each focus is a constant value.
For example, if
10, then
10
FP F P
FH F H
′
+ =
′
+ = .
Notice the foci always lie
‘inside’ the ellipse.
C
The major axis of the ellipse at the left
is the segment from
to
V V
′
, which are
called the vertices. Vertices are
always the endpoints of the major axis.
The minor axis (below left) is the
segment form
to
M M
′
. In this ellipse
the major axis is horizontal and the
minor axis is vertical.
The a value is the distance between the
center a vertex. The b value is the distance
between the center and an endpoint of the
minor axis. The c value is the distance
between the center and a focus.
Length of major axis = 2a
Length of minor axis = 2b
Length between 2 foci = 2
c
Minor Axis
