Recall that a hyperbola 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 a
hyperbola is given by the following deﬁnition.
Deﬁnition.
Let
F
1
and
F
2
be any two distinct points in the plane. A hyperbola 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
between
A
and
B
. The points
F
1
and
F
2
are called the
foci
of the hyperbola. The graphs of
hyperbolas are shown below.
In the above ﬁgure, the points
V
1
and
V
2
are called
vertices
of the hyperbola. The line containing
the foci is called the
transverse axis
, and the line orthogonal to the transverse axis through the
center of the hyperbola is called the
conjugate axis
.
The equation of a hyperbola is simpliﬁed when its transverse axis is either horizontal or vertical.
Hyperbolas 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 distance
between the vertices, the number
b
determines the slope of the hyperbola’s asymptotes, and the
number
c
is half the distance between the foci. Note: It can be seen from the diagrams that
a<c
and
c
2
=
a
2
+
b
2
.
Example.
Find the equation of a hyperbola having foci at (
±
4
,
0) and a vertex at (

2
,
0).
Solution. Since the foci are at (
±
4
,
0), the hyperbola is centered at (0
,
0) and the transverse axis is
horizontal. Hence, the equation is
x
2
a
2

y
2
b
2
= 1. Since one vertex is at
x
=

2,
a
= 2. Since the foci
are at (
±
4
,
0),
c
= 4. Since
c
2
=
a
2
+
b
2
and
c
=4
,4
2
=2
2
+
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 Fall '11
 Kutter
 Algebra, Quadratic equation, Elementary algebra, B2 Spirit

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