Section 9.3 The Ellipse
Recall that an ellipse can be obtained as the graph of the quadratic equation
Ax
2
+
Bxy
+
Cy
2
+
Dx
+
Ey
+
F
= 0 with discriminant
B
2

4
AC <
0. An alternate method for obtaining an
ellipse is given by the following deﬁnition.
Let
F
1
and
F
2
be any two distinct points in the plane. An
ellipse
is the set of all points
P
in the
plane such that
d
(
P, F
1
)+
d
(
P, F
2
) = constant, where
d
(
A, B
) denotes the distance from A to B.
The points
F
1
and
F
2
are called the
foci
of the ellipse. The graph of an ellipse is shown below.
In the above ﬁgure, the points
V
1
and
V
2
are called
vertices
of the ellipse.
The line segment
between
V
1
and
V
2
is called the
major axis
. The line segment through the center of the ellipse and
perpendicular to the major axis is called the
minor axis
.
The equation of an ellipse is much simpler when its major axis is either horizontal or vertical.
Ellipses centered at the origin and having horizontal and vertical axes are shown below. Note in
these diagrams the relationship between the distances
a
,
b
,and
c
. The number
a
is half the length
of the major axis, the number
b
is half the length of the minor axis, and the number
c
is half the
distance between the foci. Hence,
a>b
a>c
.Ineachcase
a
2
=
b
2
+
c
2
.
x
2
a
2
+
y
2
b
2
=1
y
2
a
2
+
x
2
b
2
Remark.
In the equation
x
2
r
2
+
y
2
s
2
= 1, the larger of
r, s
is the number
a
and the smaller of
r, s
is the number
b
. The larger of the two numbers
r
,
s
determines whether the major axis is either
horizontal or vertical.
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 Fall '11
 Kutter
 Algebra, Quadratic equation, Elementary algebra, eccentricity

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