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11-3 Hyperbola 7

SECTION 11-3 Hyperbola

• Definition of a Hyperbola

• Drawing a Hyperbola

• Standard Equations and Their Graphs

• Applications

As before, we start with a coordinate-free definition of a hyperbola. Using this d

nition, we show how a hyperbola can be drawn and we derive standard equations

hyperbolas specially located in a rectangular coordinate system.

The following is a coordinate-free definition of a hyperbola:•Definition of a Hyperbola

(A) If the straight-line leading edge is parallel to the major

axis of the ellipse and is 1.14 feet in front of it, and if

the leading edge is 46.0 feet long (including the width

of the fuselage), find the equation of the ellipse. Let the

x axis lie along the major axis (positive right), and let

the y axis lie along the minor axis (positive forward).

(B) How wide is the wing in the center of the fuselage (as-

suming the wing passes through the fuselage)?

Compute quantities to 3 significant digits.

42. Naval Architecture. Currently, many high-performance

racing sailboats use elliptical keels, rudders, and main sails

for the same reasons stated in Problem 41—less drag along

the trailing edge. In the accompanying figure, the ellipse

containing the keel has a 12.0-foot major axis. The straight-

line leading edge is parallel to the major axis of the ellipse

Leading edge

Fuselage Trailing edge

Ellipticalwings and tail

and 1.00 foot in front of it. The chord is 1.00 foot sho

than the major axis.

(A) Find the equation of the ellipse. Let the y

axis lie athe minor axis of the ellipse, and let the x axis lie a

the major axis, both with positive direction upwar

(B) What is the width of the keel, measured perpendic

to the major axis, 1 foot up the major axis from the

tom end of the keel?

Compute quantities to 3 significant digits.

RudderKeel

5

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800 11 Additional Topics in Analytic Geometry

DEFINITION 1 Hyperbola

A hyperbola is the set of all points P in a

plane such that the absolute value of the dif-

ference of the distances of P to two fixed

points in the plane is a positive constant.

Each of the fixed points, F and F , is calleda focus. The intersection points V and V of

the line through the foci and the two

branches of the hyperbola are called ver-

tices, and each is called a vertex. The line

segment V V is called the transverse axis.

The midpoint of the transverse axis is the

center of the hyperbola.

Thumbtacks, a straightedge, string, and a pencil are all that are needed to draw a

hyperbola (see Fig. 1). Place two thumbtacks in a piece of cardboard—

these form thefoci of the hyperbola. Rest one corner of the straightedge at the focus F so that it is

free to rotate about this point. Cut a piece of string shorter than the length of the

straightedge, and fasten one end to the straightedge corner A and the other end to the

thumbtack at F . Now push the string with a pencil up against the straightedge at B.

Keeping the string taut, rotate the straightedge about F , keeping the corner at F . The

resulting curve will be part of a hyperbola. Other parts of the hyperbola can be drawn

by changing the position of the straightedge and string. To see that the resulting curve

meets the conditions of the definition, note that the difference of the distances BF

and BF is

B

A

F Ј

String

F

FIGURE 1 Drawing a hyperbola.

ϭ Constant

ϭ Straightedge

length Ϫ String

length ϭ AF Ј Ϫ ( BF ϩ BA)

BF Ј Ϫ BF ϭ BF Ј ϩ BA Ϫ BF Ϫ BA

•Drawing a

Hyperbola

F

P

F ЈV Ј

V

d 1 Ϫ d 2 ϭ Constant

d 1

d 2

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11-3 Hyperbola 8

Using the definition of a hyperbola and the distance-between-two-points formula

can derive the standard equations for a hyperbola located in a rectangular coordi

system. We start by placing a hyperbola in the coordinate system with the foci on

x axis equidistant from the origin at F (Ϫc, 0) and F (c, 0), c Ͼ 0, as in Figure

Just as for the ellipse, it is convenient to represent the constant difference by

a Ͼ 0. Also, the geometric fact that the difference of two sides of a triangle is alw

less than the third side can be applied to Figure 2 to derive the following useful re

We will use this result in the derivation of the equation of a hyperbola, which we n

begin.

Referring to Figure 2, the point P( x, y) is on the hyperbola if and only if

After eliminating radicals and absolute value signs by appropriate use of squaring

simplifying, another good exercise for you, we have

Dividing both sides of equation (2) by a2(c2 Ϫ a2) is permitted, since neither a2

c2

Ϫ a2

is 0. From equation (1), a Ͻ c; thus, a2

Ͻ c2

and c2

Ϫ a2

Ͼ 0. The consa was chosen positive at the beginning.

To simplify equation (3) further, we let

b2 ϭ c2 Ϫ a2 b Ͼ 0

to obtain

From equation (5) we see that the x intercepts, which are also the vertices,

x ϭ Ϯa and there are no y intercepts. To see why there are no y intercepts x ϭ 0 and solve for y:

An imaginary number y ϭ Ϯ Ϫb2

y2 ϭ Ϫb2

02

a2Ϫ

y2

b2ϭ 1

x2

a2

y2

b21

x2

a2Ϫ

y2

c2 Ϫ a2ϭ 1

(c2 Ϫ a2) x2 Ϫ a2 y2 ϭ a2(c2 Ϫ a2)

Խ ( x ϩ c)2 ϩ y2 Ϫ ( x Ϫ c)2 ϩ y2Խ ϭ 2a Խd (P, F

Ј)

Ϫd (P, F )Խ

ϭ2a

Խd 1 Ϫ d 2Խ ϭ 2a

a Ͻ c

2a Ͻ 2c

Խd 1 Ϫ d 2Խ Ͻ 2c

•Standard Equationsand Their Graphs

FIGURE 2 Hyperbola with foci onthe x axis.

d 1 d 2

x

y

F Ј(Ϫc , 0) F (c , 0)

c Ͼ 0 d 1 Ϫ d 2 ϭ Positive constant

P (x , y )

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If we start with the foci on the y axis at F (0, Ϫc) and F (0, c) as in Figure 3,

instead of on the x axis as in Figure 2, then, following arguments similar to those

used for the first derivation, we obtain

(6)

where the relationship among a, b, and c remains the same as before:

b2 ϭ c2 Ϫ a2 (7)

The center is still at the origin, but the transverse axis is now on the y axis.

As an aid to graphing equation (5), we solve the equation for y in terms of x,

another good exercise for you, to obtain

(8)

As x changes so that becomes larger, the expression 1 Ϫ (a2 / x2) within the radi-

cal approaches 1. Hence, for large values of , equation (5) behaves very much like

the lines

(9)

These lines are asymptotes for the graph of equation (5). The hyperbola approaches

these lines as a point P( x, y) on the hyperbola moves away from the origin (see Fig.

4). An easy way to draw the asymptotes is to first draw the rectangle as in Figure 4,

then extend the diagonals. We refer to this rectangle as the asymptote rectangle.

Starting with equation (6) and proceeding as we did for equation (5), we obtain

the asymptotes for the graph of equation (6):

(10)

The perpendicular bisector of the transverse axis, extending from one side of the

asymptote rectangle to the other, is called the conjugate axis of the hyperbola.

Given an equation of the form (5) or (6), how can we find the coordinates of the

foci without memorizing or looking up the relation b2 ϭ c2 Ϫ a2? Just as with the

y ϭ Ϯa

b x

x

y

b

Ϫa a 0

Ϫb

Asymptoteb

a x y ϭ Ϫ

Asymptoteb

a x y ϭ

x 2 y 2

a 2 b 2Ϫ ϭ 1

FIGURE 4 Asymptotes.

y ϭ Ϯb

a x

Խ xԽԽ x

Խ

y ϭ Ϯb

a x 1 Ϫ

a2

x2

y2

a2

x2

b21

FIGURE 3 Hyperbola with foci onthe y axis.

x

y

F Ј(0, Ϫc )

d 1

d 2

F (0, c )

P (x , y )

c Ͼ 0 d 1 Ϫ d 2 ϭ Positive constant

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11-3 Hyperbola 8

ellipse, there is a simple geometric relationship in a hyperbola that enables us to

the same result using the Pythagorean theorem. To see this relationship, we rew

b2 ϭ c2 Ϫ a2 in the form

c2 ϭ a2 ϩ b2

Note in the figures in Theorem 1 below that the distance from the center to a fo

is the same as the distance from the center to a corner of the asymptote rectanStated in another way:

A circle, with center at the origin, that passes through all four corners

of the asymptote rectangle also passes through all foci of hyperbolas with

asymptotes determined by the diagonals of the rectangle.

We summarize all the preceding results in Theorem 1 for convenient referen

Theorem 1 Standard Equations of a Hyperbola with Center at (0, 0)

1.

x intercepts: Ϯa (vertices)

y intercepts: none

Foci: F (Ϫc, 0), F (c, 0)

c2 ϭ a2 ϩ b2

Transverse axis length ϭ 2a

Conjugate axis length ϭ 2b

2.

x intercepts: none

y intercepts: Ϯa (vertices)

Foci: F (0, Ϫc), F (0, c)

c2 ϭ a2 ϩ b2

Transverse axis length ϭ 2a

Conjugate axis length ϭ 2b

[ Note: Both graphs are symmetric with respect to the x axis, y axis, and origin

EXPLORE-DISCUSS 1 The line through a focus F of a hyperbola that is perpendicular to the transvers

axis intersects the hyperbola in two points G and H . For each of the two standar

equations of a hyperbola with center (0, 0), find an expression in terms of a an

b for the distance from G to H .

a

c

Ϫb b

c

Ϫc

Ϫa

F

F Ј

x

y

y2

a2Ϫ

x2

b2ϭ 1

b

c

Ϫa a Ϫc c

Ϫb

F F Јx

y

x2

a2Ϫ

y2

b2ϭ 1

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EXAMPLE 1 Graphing Hyperbolas

Sketch the graph of each equation, find the coordinates of the foci, and find the lengths

of the transverse and conjugate axes.

(A) 9 x2 Ϫ 16 y2 ϭ 144 (B) 16 y2 Ϫ 9 x2 ϭ 144 (C) 2 x2 Ϫ y2 ϭ 10

Solutions (A) First, write the equation in standard form by dividing both sides by 144:

a 2 ϭ 16 and b 2 ϭ 9

Locate x intercepts, x ϭ Ϯ4; there are no y intercepts. Sketch the asymptotes

using the asymptote rectangle, then sketch in the hyperbola (Fig. 5).

Thus, the foci are F (Ϫ5, 0) and F (5, 0)

Transverse axis length ϭ 2(4) ϭ 8

Conjugate axis length ϭ 2(3) ϭ 6

(B)

a 2 ϭ 9 and b 2 ϭ 16

Locate y intercepts, y ϭ Ϯ3; there are no x intercepts. Sketch the asymptotes

using the asymptote rectangle, then sketch in the hyperbola (Fig. 6). It is impor-

tant to note that the transverse axis and the foci are on the y axis.

Thus, the foci are F (0, Ϫ5) and F (0, 5).

Transverse axis length ϭ 2(3) ϭ 6

Conjugate axis length ϭ 2(4) ϭ 8

c ϭ 5

ϭ 25

ϭ 9 ϩ 16

Foci: c2 ϭ a2ϩ b2

y2

9Ϫ

x2

16ϭ 1

16 y2 Ϫ 9 x2 ϭ 144

c ϭ 5

ϭ 25

ϭ 16 ϩ 9

Foci: c2 ϭ a2ϩ b2

x2

16Ϫ

y2

9ϭ 1

9 x2 Ϫ 16 y2 ϭ 144

FIGURE 5 9 x2 Ϫ 16 y2 ϭ 144.

c Ϫc

c

F F Јx

y

Ϫ5

6Ϫ6

5

FIGURE 6 16 y2 Ϫ 9 x2 ϭ 144.

c

c

Ϫc

F

F Ј

x

y

Ϫ6

6Ϫ6

6

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11-3 Hyperbola 8

(C)

a 2 ϭ 5 and b 2 ϭ 10

Locate x intercepts, x ϭ ; there are no y intercepts. Sketch the asympt

using the asymptote rectangle, then sketch in the hyperbola (Fig. 7).

Thus, the foci are F ( ) and F ( ).

Remark. To graph the equation 9 x2 Ϫ 16 y2 ϭ 144 of Example 1A on a graph

utility we first solve the equation for y, obtaining . We then gr

each of the two functions. The graph of is the upper half of

hyperbola, and the graph of is the lower half.

Matched Problem 1 Sketch the graph of each equation, find the coordinates of the foci, and find the len

of the transverse and conjugate axes.

(A) 16 x2 Ϫ 25 y2 ϭ 400 (B) 25 y2 Ϫ 16 x2 ϭ 400 (C) y2 Ϫ 3 x2 ϭ 12

Hyperbolas of the form

and M , N Ͼ 0

are called conjugate hyperbolas. In Example 1 and Matched Problem 1, the hybolas in parts A and B are conjugate hyperbolas—they share the same asymptot

CAUTION When making a quick sketch of a hyperbola, it is a common error to have th

hyperbola opening up and down when it should open left and right, or vice

versa. The mistake can be avoided if you first locate the intercepts accurately

y2

N Ϫ

x2

M ϭ 1

x2

M Ϫ

y2

N ϭ 1

y ϭ Ϫ 9 x2 Ϫ 144

16

y ϭ 9 x2 Ϫ 144

16

y ϭ Ϯ 9 x2 Ϫ 144

16

Conjugate axis length ϭ 2 10 Ϸ 6.32

Transverse axis length ϭ 2 5 Ϸ 4.47

15, 0Ϫ 15, 0

c ϭ 15

ϭ 15

ϭ 5 ϩ 10

Foci: c2ϭ a2

ϩ b2

Ϯ 5

x2

5Ϫ

y2

10ϭ 1

2 x2Ϫ y2 ϭ 10

FIGURE 7 2 x2 Ϫ y2 ϭ 10.

Ϫc c

c

F Ј F x

y

Ϫ5

5Ϫ5

5

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EXAMPLE 2 Finding the Equation of a Hyperbola

Find an equation of a hyperbola in the form

M , N Ͼ 0

if the center is at the origin, and:

(A) Length of transverse axis is 12 (B) Length of transverse axis is 6

Length of conjugate axis is 20 Distance of foci from center is 5

Solutions (A) Start with

and find a and b:

Thus, the equation is

(B) Start with

and find a and b:

To find b, sketch the asymptote rectangle (Fig. 8), label known parts, and use

the Pythagorean theorem:

Thus, the equation is

y2

9Ϫ

x2

16ϭ 1

b ϭ 4

ϭ 16

b2ϭ 52

Ϫ 32

a ϭ6

2ϭ 3

y2

a2Ϫ x

2

b2ϭ 1

y2

36Ϫ

x2

100ϭ 1

a ϭ12

2ϭ 6 and b ϭ

20

2ϭ 10

y2

a2Ϫ

x2

b2ϭ 1

y2

M Ϫ

x2

N ϭ 1

FIGURE 8 Asymptote rectangle.

b Ϫb

5 3

F Ј

F

x

y

Ϫ5

5

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11-3 Hyperbola 8

Matched Problem 2 Find an equation of a hyperbola in the form

M , N Ͼ 0


(A) Length of transverse axis is 50 (B) Length of conjugate axis is 12

Length of conjugate axis is 30 Distance of foci from center is

EXPLORE-DISCUSS 2 (A) Does the line with equation y ϭ x intersect the hyperbola with equatio

x2 Ϫ ( y2 /4) ϭ 1? If so, find the coordinates of all intersection points.

(B) Does the line with equation y ϭ 3 x intersect the hyperbola with equatio

x2 Ϫ ( y2 /4) ϭ 1? If so, find the coordinates of all intersection points.

(C) For which values of m does the line with equation y ϭ mx intersect the hyper

bola ? Find the coordinates of all intersection points.

• Applications You may not be aware of the many important uses of hyperbolic forms. They

encountered in the study of comets; the loran system of navigation for pleasure bo

ships, and aircraft; sundials; capillary action; nuclear cooling towers; optical

radiotelescopes; and contemporary architectural structures. The TWA buildin

Kennedy Airport is a hyperbolic paraboloid, and the St. Louis Science Center P

etarium is a hyperboloid. (See Fig. 9.)

Comet around sun Loran navigation St. Louis planetarium(a) (b) (c)

FIGURE 9 Uses of hyperbolic forms.

Some comets from outer space occasionally enter the sun’s gravitational fi

follow a hyperbolic path around the sun (with the sun at a focus), and then le

Ship

S 1S 2

S 3

p 1 p 2

q 1

q 2

Sun

Comet

x2

a2Ϫ

y2

b2ϭ 1

x2

M Ϫ

y2

N ϭ 1

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never to be seen again [Fig. 9(a)]. In the loran system of navigation, transmitting sta-

tions in three locations, S 1, S 2, and S 3 [Fig. 9(b)], send out signals simultaneously. A

ship with a receiver records the difference in the arrival times of the signals from S 1and S 2 and also records the difference in arrival times of the signals from S 2 and S 3.

The difference in arrival times can be transformed into differences of the distances

that the ship is to S 1 and S 2 and to S 2 and S 3. Plotting all points so that these differ-

ences in distances remain constant produces two branches, p1 and p2, of a hyperbola

with foci S 1 and S 2 and two branches, q1 and q2, of a hyperbola with foci S 2 and S 3.It is easy to tell which branches the ship is on by noting the arrival times of the sig-

nals from each station. The intersection of a branch from each hyperbola locates the

ship. Most of these calculations are now done by shipboard computers, and positions

in longitude and latitude are given. This system of navigation is widely used for

coastal navigation. Inexpensive loran units are now found on many small pleasure

boats. Figure 9(c) illustrates a hyperboloid used architecturally. With such structures,

thin concrete shells can span large spaces.

Answers to Matched Problems

1. (A)

(B)

(C)

2. (A) (B) x2

45Ϫ

y2

36ϭ 1

x2

625Ϫ

y2

225ϭ 1

c

c

Ϫc F Ј

F

y 2 x 2

12 4Ϫ ϭ 1

Foci: F Ј(0, Ϫ4), F (0, 4) Transverse axis length ϭ 2 12Ϸ 6.93Conjugate axis length ϭ 4

x

y

Ϫ6

5Ϫ

5

6

c

c

Ϫc

F

F Ј

y 2 x 2

16 25Ϫ ϭ 1

Foci: F Ј(0, Ϫ 41), F (0, 41) Transverse axis length ϭ 8Conjugate axis length ϭ 10

x

y

Ϫ10

10Ϫ10

10

c Ϫc

c F Ј F

x 2 y 2

25 16Ϫ ϭ 1

Foci: F Ј(Ϫ 41, 0), F ( 41, 0) Transverse axis length ϭ 10Conjugate axis length ϭ 8

x

y

Ϫ10

10Ϫ10

10

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11-3 Hyperbola 8

EXERCISE 11-3

A

Sketch a graph of each equation in Problems 1–8, find the

coordinates of the foci, and find the lengths of the transverse

and conjugate axes.

1. 2.

3. 4.

5. 4 x2 Ϫ y2 ϭ 16 6. x2 Ϫ 9 y2 ϭ 9

7. 9 y2 Ϫ 16 x2 ϭ 144 8. 4 y2 Ϫ 25 x2 ϭ 100

BSketch a graph of each equation in Problems 9–12, find the

coordinates of the foci, and find the lengths of the transverse

and conjugate axes.

9. 3 x2 Ϫ 2 y2 ϭ 12 10. 3 x2 Ϫ 4 y2 ϭ 24

11. 7 y2 Ϫ 4 x2 ϭ 28 12. 3 y2 Ϫ 2 x2 ϭ 24

In Problems 13–18, find an equation of a hyperbola in the

form


13. Transverse axis on y axis

Transverse axis length ϭ 10

Conjugate axis length ϭ 18

14. Transverse axis on x axis



15. Transverse axis on x axis


Distance of foci from center ϭ 9

16. Transverse axis on y axis


Distance of foci from center ϭ 25

17. Conjugate axis on x axis


Distance between foci ϭ 12

18. Conjugate axis on y axis


Distance between foci ϭ 48

2

x2

M

Ϫ y2

N

ϭ 1 or y2

N

Ϫ x2

M

ϭ 1 M , N Ͼ 0

y2

25Ϫ

x2

9ϭ 1

y2

4Ϫ

x2

9ϭ 1

x2

9Ϫ

y2

25ϭ 1

x2

9Ϫ

y2

4ϭ 1

19. (A) How many hyperbolas have center at (0, 0) and a f

at (1, 0)? Find their equations.

(B) How many ellipses have center at (0, 0) and a foc

(1, 0)? Find their equations.

(C) How many parabolas have vertex at (0, 0) and foc

(1, 0)? Find their equations.

20. How many hyperbolas have the lines y ϭ Ϯ2 x as asy

totes? Find their equations.

In Problems 21–24, graph each system of equations in th

same rectangular coordinate system and find the coordin

of any points of intersection.

Check Problems 21–24 with a graphing utility.

21. 3 y2 Ϫ 4 x2 ϭ 12 22. y2 Ϫ x2 ϭ 3

y2 ϩ x2 ϭ 25 y2 ϩ x2 ϭ 5

23. 2 x2

ϩ y2

ϭ 24 24. 2 x2

ϩ y2

ϭ 17 x2 Ϫ y2 ϭ Ϫ12 x2 Ϫ y2 ϭ Ϫ5

In Problems 25–28, find all points of intersection for eac

system of equations to three decimal places.

Check Problems 25–28 with a graphing utility.

25. y2 Ϫ x2 ϭ 9 26. y2 Ϫ x2 ϭ 4

2 y Ϫ x ϭ 8 y Ϫ x ϭ 6

y Ն 0 y Ն 0

27. y2 Ϫ x2 ϭ 4 28. y2 Ϫ x2 ϭ 1

y2 ϩ 2 x2 ϭ 36 2 y2 ϩ x2 ϭ 16

In Problems 29–

32, determine whether the statement is tor false. If true, explain why. If false, give a counterexam

29. The line segment joining the foci of a hyperbola has gr

length than the conjugate axis.

30. The line segment joining the foci of a hyperbola has gr

length than the transverse axis.

31. Every line through the center of 4 x2 Ϫ y2 ϭ 16 inter

the hyperbola in exactly two points.

32. Every nonvertical line through a vertex of 4 x2 Ϫ y2 ϭ 1

tersects the hyperbola in exactly two points.

C

Eccentricity. The set of points in a plane each of whose di

tance from a fi xed point is e times its distance from a fi xed

is a conic section. The positive number e is called the ecce

tricity of the conic section. Problems 33 and 34 below and

Problems 33 and 34 in Section 12-2 illustrate an approach

de fining the conic sections in terms of eccentricity.

33. Find an equation of the set of points in a plane eac

whose distance from (3, 0) is three-halves its distance f

the line x ϭ . Identify the geometric figure.43

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34. Find an equation of the set of points in a plane each of

whose distance from (0, 4) is four-thirds its distance from

the line y ϭ . Identify the geometric figure.

In Problems 35–38, use a graphing utility to find the coordi-

nates of all points of intersection to two decimal places.

35. 2 x2 Ϫ 3 y2 ϭ 20, 7 x ϩ 15 y ϭ 10

36. y2 Ϫ3 x2 ϭ 8, x2 ϭ

37. 24 y2 Ϫ 18 x2 ϭ 175, 90 x2 ϩ 3 y2 ϭ 200

38. 8 x2 Ϫ 7 y2 ϭ 58, 4 y2 Ϫ 11 x2 ϭ 45

APPLICATIONS

39. Architecture.An architect is interested in designing a thin-

shelled dome in the shape of a hyperbolic paraboloid, as

shown infi

gure (a). Find the equation of the hyperbola lo-cated in a coordinate system [Fig. (b)] satisfying the indi-

cated conditions. How far is the hyperbola above the vertex

6 feet to the right of the vertex? Compute the answer to two

decimal places.

Hyperbolic paraboloid(a)

Hyperbola part of dome(b)

40. Nuclear Power. A nuclear cooling tower is a hyperboloid,

that is, a hyperbola rotated around its conjugate axis, as

shown in Figure (a). The equation of the hyperbola in Fig-

ure (b) used to generate the hyperboloid is

x2

1002Ϫ

y2

1502ϭ 1

(8, 12)

x

y

10Ϫ10

10

Hyperbola

Parabola

Ϫ y3

94

Nuclear cooling tower(a)

Hyperbola part of dome(b)

If the tower is 500 feet tall, the top is 150 feet above the

center of the hyperbola, and the base is 350 feet below the

center, what is the radius of the top and the base? What is

the radius of the smallest circular cross section in the

tower? Compute answers to 3 significant digits.

41. Space Science. In tracking space probes to the outer planets,NASAuses large parabolic reflectors with diameters equal to

two-thirds the length of a football field. Needless to say,

many design problems are created by the weight of these re-

flectors. One weight problem is solved by using a hyperbolic

reflector sharing the parabola’s focus to reflect the incoming

electromagnetic waves to the other focus of the hyperbola

where receiving equipment is installed (see the figure).

(a)

F Ј

F

Incomingwave

Commonfocus

Hyperbola

Hyperbolafocus

Receiving coneParabola

x

y

Ϫ500

500Ϫ500

500

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11-4 Translation of Axes 8

SECTION 11-4 Translation of Axes

• Translation of Axes

• Standard Equations of Translated Conics

• Graphing Equations of the Form Ax2 ϩ Cy2 ϩ Dx ϩ Ey ϩ F ϭ 0

• Finding Equations of Conics

In the last three sections we found standard equations for parabolas, ellipses,

hyperbolas located with their axes on the coordinate axes and centered relative to

origin. What happens if we move conics away from the origin while keeping t

axes parallel to the coordinate axes? We will show that we can obtain new stand

equations that are special cases of the equation Ax2 ϩ Cy2 ϩ Dx ϩ Ey ϩ F ϭ

where A and C are not both zero. The basic mathematical tool used in this endea

is translation of axes. The usefulness of translation of axes is not limited to graph

conics, however. Translation of axes can be put to good use in many other graphsituations.

A translation of coordinate axes occurs when the new coordinate axes have the s

direction as and are parallel to the original coordinate axes. To see how coordin

in the original system are changed when moving to the translated system, and v

versa, refer to Figure 1.

x Ј

y Ј

x

y

0

P (x , y )P (x Ј, y Ј)

(0Ј, 0Ј)

(0, 0)

(h , k )

0Ј

y Јy

x Ј

x

FIGURE 1 Translation of coordinates.

•Translation of Axes

(b)

Radiotelescope

For the receiving antenna shown in the figure, the com

focus F is located 120 feet above the vertex of the parab

and focus F (for the hyperbola) is 20 feet above the ve

The vertex of the reflecting hyperbola is 110 feet abov

vertex for the parabola. Introduce a coordinate system

using the axis of the parabola as the y axis (up positive)

let the x axis pass through the center of the hyperbola (

positive). What is the equation of the reflecting hyperb

Write y in terms of x.

bpc5_ch11-03

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