Section 10.4
The Hyperbola
OBJECTIVE 1
Find an equation of the hyperbola with center at the origin, one
focus at (– 5, 0), and one vertex at (2, 0). Graph the equation.
2
2
x
y
Use a graphing utility to graph the ellipse

1
36 25
x2 y 2
Analyze the equation

1
25 16
Analyze the equation 2 y  8x  32
2
2
2
2
2
y

8
x
32
a  16, b  4

2
2
2
32
32
b  c a
2
2
2
y
x
4  c  16

1
16 4
c 2  20
2
2
a  4, b  2, c  20  2 5
Transverse axis parallel to y-axis
vertices: (0, 4), (0,4)
foci : (0,  20), (0, 20)
Find an equation of the hyperbola having one vertex at (0, 4) and
a4
foci at (0, – 7) and (0, 7). Graph the equation.
c7
Foci and vertices are parallel to y-axis
b2  c 2  a 2
b2  72  42  49  16  33
y2 x2
 2 1
2
a
b
y2 x2

1
16 33
OBJECTIVE 2
Analyze the equation 4 x  y  16
2
2
4 x 2  y 2 16
a  4, b  16

16
16
b2  c 2  a 2
2
2
2
4
x
y
16  c  4

1
2
16 16
c  20
2
2
x
y
a  2, b  4, c  20  2 5

1
4 16
Transverse axis parallel to x-axis
Vertices: ( 2,0),(2,0)
b 4
Asymptotes:
 2
a 2
Foci: ( 2 5,0),(2 5,0)
y  2 x , y  2 x
2
2
OBJECTIVE 3
Find an equation for the hyperbola with center at (–1, 3), one
focus at (– 1, – 1), and one vertex at (– 1, 4). Graph the
equation by hand.
a  1, c  4
h  1, k  3
b2  c 2  a 2  16  1  15
Transverse axis parallel to y-axis
( y  k )2 ( x  h)2

1
2
2
a
b
( y  3) 2 ( x  1) 2

1
1
15
Find an equation for the hyperbola with center at (–2, 3), one
focus at (2, 3), and one vertex at (0, 3). Graph the equation by
hand.
a  2, c  4
h  2, k  3
b2  c 2  a 2  16  4  12
Transverse axis parallel to x-axis
( x  k )2 ( y  h)2

1
2
2
a
b
( x  2) 2 ( y  3) 2

1
4
12
Analyze the equation  16 x  y  64x  6 y  71  0
2
Center: ( 2,3)
h  2, k  3, a  4, b  1
Vertices:
( 2,3  4),( 2,3  4)
2
16 x 2  64 x  y 2  6 y  71
16( x 2  4 x)  y 2  6 y  71
16( x 2  4 x  4)  y 2  6 y  9  71  64  9
2
2

16(
x

2)

(
y

3)
 16
 ( 1,3),(7,3)
2
( y  3)
2

(
x

2)

1
Foci:
16
c 2  a 2  b2 =16+1=17
( x  2) 2 ( y  3) 2

1
1
16
( 2,3  17),( 2,3  17)
2
2
(
y

3)
(
x

2)
 ( 2,7.123),( 2, 1.123)

1
16
1
Analyze the equation  16 x  y  64x  6 y  71  0
2
Center: ( 2,3)
h  2, k  3, a  4, b  1
Vertices:
( 2,3  4),( 2,3  4)
 ( 1,3),(7,3)
Foci:
c 2  a 2  b2 =16+1=17
( 2,3  17),( 2,3  17)
 ( 2,7.123),( 2, 1.123)
2
( y  3) 2 ( x  2) 2

1
16
1
a 4
Asymptotes: Slope=   4
b 1
a
y  k  ( x  h)
b
y  3  4( x  2)
y  4( x  2)  3  4 x  8  3  4 x  11
a
y  h   (x  k)
b
y  3  4( x  2)
y  4( x  2)  3  4 x  8  3  4 x  5
OBJECTIVE 4