Section 10.4
Conic Sections Hyperbolas
Hyperbola
Hyperbola – is the set of
points (P) in a plane such
that the difference of the
distances from P to two
fixed points F1 and F2 is a
given constant k.
PF1  PF2  k
P
F1
F2
Hyperbola
Asymptotes
b
y x
a
b
y x
a
Transverse
Axis
F1
F2
Vertices = (a, 0)
Hyperbola - Equation
For a hyperbola with a
horizontal transverse axis,
the standard form of the
equation is:
2
P
2
x
y
 2 1
2
a
b
F1
F2
Hyperbola
a
y x
b
a
y x
b
F1
F2
Transverse
Axis
Hyperbola - Equation
For a hyperbola with a
vertical transverse axis, the
standard form of the
equation is:
2
F2
2
y
x
 2 1
2
b
a
F1
Hyperbola
Definitions:
• a – is the distance between the vertex and the center of
the hyperbola
• b – is the distance between the tangent to the vertex and
where it intersects the asymptotes
• c – is the distance between the foci and the center
Relationships:
The distances a, b and c form a right triangle and can be used
to construct the hyperbola.
Horizontal_Hyperbola.html
Vertical_Hyperbola.html
Find the Foci
2
2
x
y

1
Find the foci for a hyperbola:
25 9
a2
b2
From the form, we know it’s a horizontal transverse
axis. We know the foci are at (c, o ) and that
c2 = a2 + b2
c  25  9


Foci are  34, 0
  34
Find the Foci
2
2
y
x

1
Find the foci for a hyperbola:
49 25
b2
a2
From the form, we know it’s a vertical transverse axis.
We know the foci are at (0, c ) and that
c2 = a2 + b2
c  49  25

Foci are 0,  74

  74
Write the Equation
Write the equation of the hyperbola with foci at
(5, 0) and vertices at (3, 0)
c
a
From the info, it’s a horizontal transversal.
We need to find b
5 2  32  b 2
25  9  b 2
16  b 2
4b
x2 y2

1
9 16
Write the Equation
Write the equation of the hyperbola with foci at
(0, 13) and vertices at (0, 5)
c
b
From the info, it’s a vertical transversal.
We need to find a
2
13  a  5
2
2
2
169  a  25
2
144  a
2
2
y
x

1
25 144
Assignment