Conic Sections
Hyperbolas
A hyperbola is the set of all points such that the difference of the distances between any point on the hyperbola and two fixed points is constant. The two fixed points are called the foci of the hyperbola.
The graph of a hyperbola is not continuous--every hyperbola has two distinct branches. The line segment containing both foci of a hyperbola whose endpoints are both on the hyperbola is called the transverse axis. The endpoints of the transverse axis are called the vertices of the hyperbola. The point halfway between the foci (the midpoint of the transverse axis) is the center.
The standard equation for a hyperbola with a horizontal transverse axis is
- 
= 1
. The center is at
(h, k)
.
The distance between the vertices is
2a
. The distance between the foci is
2c
.
c
2 = a
2 + b
2
. The line segment of length
2b
perpendicular to the
transverse axis whose midpoint is the center is the conjugate axis of the
hyperbola.
The standard equation for a hyperbola with a vertical transverse axis is
- 
= 1
. The center is at
(h, k)
.
The distance between the vertices is
2a
. The distance between the foci is
2c
.
c
2 = a
2 + b
2
.
Every hyperbola has two asymptotes.
A hyperbola with a horizontal transverse axis and center at
(h, k)
has one
asymptote with equation
y = k + 
(x - h)
and the other with equation
y = k - (x - h)
. A hyperbola with a vertical transverse axis and
center at
(h, k)
has one asymptote with equation
y = k + 
(x - h)
and the other with equation
y = k - (x - h)
.
The eccentricity of a hyperbola, like an ellipse, is
e = 
. For
all hyperbolas, though,
c > a
, so
e > 1
. If
e
is close to one, the
branches of the hyperbola are very narrow, but if
e
is much greater than one,
then the branches of the hyperbola are very flat.
