11.2 HyperbolasObjectives:

1. Define a hyperbola2. Write the equation of a hyperbola3. Identify important characteristics of

hyperbolas4. Graph hyperbolas

HyperbolaThe set of all points for which the difference

of the distances from two points is constant.

Equation of a Hyperbola Centered on the Origin

Characteristics of a HyperbolaImportant

Facts:

The hyperbola bends toward the foci

The positive term determines which way the hyperbola opens

The distance between the foci is 2c

The distance between the vertices is 2a

The center is the midpoint between the foci and the midpoint between the vertices

c2 = a2 + b2

Example #1Show that the graph of the equation is a

hyperbola. Graph it and its asymptotes. Find the equations of the asymptotes, and label the foci and the vertices.64164 22 xy

124

1416

64

64

64

16

64

4

2

2

2

2

22

22

xy

xy

xy

5220

416

24

2,4

22

c

c

c

ba

xy

xy

xb

ay

Asymptotes

22

4

:

Example #1Show that the graph of the equation is a

hyperbola. Graph it and its asymptotes. Find the equations of the asymptotes, and label the foci and the vertices.64164 22 xy

124

:

2

2

2

2

xy

Equation

xy

Asymptotes

2

:

52,0:

Foci

4,0:

Vertices

1 2 3 4 5 6 7 8 9 10–1–2–3–4–5–6–7–8–9–10 x

12345678910

–1–2–3–4–5–6–7–8–9–10

y

Example #2Graph the following hyperbola using a

graphing calculator.

2045 22 yx

4

520

4

520

5204

2

22

22

xy

xy

xy

Example #3AA. Find the equation of the hyperbola that has

vertices at (2, 0) and (-2, 0) and passes through Then sketch its graph by using the asymptotes, and label the foci.

3,4

13

4

16

13

2

4

12

2

2

2

2

2

2

2

2

2

b

b

b

yx

1

33

33

13

4

2

2

2

b

b

b

b

With the vertices on the x-axis, this implies a = 2.

5

14

12 22

c

14

22

yx

xy

xa

by

2

1

Example #3AA. Find the equation of the hyperbola that has

vertices at (2, 0) and (-2, 0) and passes through Then sketch its graph by using the asymptotes, and label the foci.

3,4

14

:

22

yx

Equation

xy

Asymptotes

2

1

:

0,5:

Foci

1 2 3 4 5 6–1–2–3–4–5–6 x

1

2

3

4

5

6

–1

–2

–3

–4

–5

–6

y

Example #3BB. Find the equation of a hyperbola with y-

intercepts at

±7 and an asymptote at xy3

1

With it intersecting the y-axis, this implies that a = 7. From the equation of the asymptote we get:

213

17

bbb

a1

217 2

2

2

2

xy

Example #3CC. Find the equation of a hyperbola with foci at

(±8, 0) and a vertex at 0,25

14

14

5064

258

8,25

2

2

22

b

b

b

b

ca

1

5064

1258

22

2

2

2

2

yx

yx

Example #4An airplane crashed and was heard by a park

ranger and by a family camping in a park. The park ranger and the family are ¼ mile apart and the ranger heard the sound 1 second before the family. The speed of sound in air is approximately 1100 feet per second. Describe the possible locations of the plane crash.The family and the ranger are placed at opposite foci of a hyperbolic curve. The crash occurred closer to the ranger than the family so the crash occurred on the branch of the hyperbola closest to the ranger.Since sound travels at 1100 ft/sec, after 1 sec it will have traveled 1100 ft. This implies the crash was 1100 ft closer to the ranger than the family, which also means the vertices are 1100 ft apart. Since 1 mile has 5280 ft, ¼ a mile is 5280 ÷ 4 = 1320 ft, which is the distance between the foci.

Example #4Describe the possible locations of the plane

crash.Distance between foci: 1320 ftDistance between vertices: 1100 ft

660

13202

550

11002

c

c

a

a

365

100,133

500,302600,435

550660

2

2

22

b

b

b

b

1365550 2

2

2

2

yx

Ranger Family

200 400 600 800 10001200–200–400–600–800–1000–1200 x

200

400

600

800

1000

1200

–200

–400

–600

–800

–1000

–1200

y

The crash occurred somewhere on the left branch of the hyperbola.