SOLUTIONWeuse a graphing device to produce the graphs for the cases *a* = -2, -1,
-0.5, -0.2,0,0.5, 1, and 2 shown inFigure 17.Notice that allof these curves (except
the case *a* =0) have two branches, and both branches approach the vertical asymptote
*x* =*a as x* approaches *a from the* left or right.

When *a*

### <

^{-1,}both branches are smooth; but when

*a*reaches - I,the right branch acquires asharp point, called a

*cusp.*

*For a*between -1 and 0the cusp turns into aloop, which becomes larger as

*a approaches*O.When

*a*=0, both branches come together and form a circle (see Example 2). For

*a*between 0and I, the left branch has a loop, which shrinks tobecome acusp when

*a*

### =

1.For*a*

### >

1,the branches become smooth again, and as*a*increases further, they become less curved. Notice that the curves with

*a*posi- tive are reflections about the y-axis of the corresponding curves with

*a*negative.

These curves are called conchoids of Nicomedes after the ancient Greek scholar Nicomedes. He called them conchoids because the shape of their outer branches

resembles that of aconch shell ormussel shell. 0

**FIGURE** **17** Members ofthe family
*x*

### =

*a*

### +

cos*f,*

*Y*

### =

*a*tan

*t*

### +

sin*f,*

allgraphed in the viewing rectangle [-4,4] by[-4,4]

### -8

^{EXERCISES}

**1-4** Sketch the curve byusing the parametric equations toplot
points. Indicate with anarrow the direction inwhich the curve is
traced as tincreases.

**5-10**

(a) Sketch the curve by using the parametric equations to plot
points. Indicate with anarrow the direction inwhich the curve
is traced as*t increases.*

(b) Eliminate the parameter to find aCartesian equation of the curve.

*5.* *x* = *3t - 5,* *Y= 2t*

### +

1*6.* *x* = 1

### +

*t,*

*Y*= 5-

*2t,*-2 ~

*t~ 3*

*7.x*=

*t*

*2*

*-*2,

*y*=5 -

*2t,*-3

*~ t ~ 4*

*8.* *x*

### =

I### +

*3t,*

*Y*

### =

2-*t*

*2*

### 00

*X*=

*jt,*

*y*

^{=}I -

*t*

*10.*

*x*

### =

*t*

*2,*

*Y*

### =

*t*

*J*

**11-18**

(a) Eliminate the parameter to find a Cartesian equation ofthe curve.

(b) Sketch the curve and indicate with anarrow the direction in which the curve istraced asthe parameter increases.

*II.* *x* = sin e, *y* = cos*e,* 0~ *e* *~* ^{7T}

12. x=4cose, y=5sine, *-7T/2~e~7T/2*

### [QJ

*x*= sin

*t,*

*y*= csc t, 0

### <

*t*

### <

*7T/2*

*14.x*= sec e,

*y = tan*

*e,*

*-7T/2*

### <

*e*

### <

*7T/2*

*15.x=e',*

*y=e-'*

*16.* *x* ^{=}In *t,* *y* ^{=}

*jt,*

*t~*I

*17.*

*x = sinh*

^{f,}*y*= cosh

^{f}**19-22** Describe the motion of aparticle with position (x, y) as
*t*varies inthe given interval.

**19.***x* =cos *7Tt,* *Y*=sin *7Tt,* I~ *t ~ 2*

23. Suppose a curve is given by the parametric equations *x* =*f(t),*
*y* =*g(t),* where the range of f is [1,4] and the range of*g*is
[2,3]. What can you say about the curve?

**24.** Match the graphs ofthe parametric equations x =*f(t)* and
*y* =*g(t)* in (a)-(d) with theparametric curves labeled I-IV.

Give reasons for your choices.

**28. Match the** parametric equations with the graphs labeled I-VI.

*x* Give reasons for your choices. (Do not usea graphing
device.)

*(a) x*= *t**4* *-* *t*

### +

I,*y*=

*t2*(b)

*x*=

*t2*

*-*

*2t,*

*y*=

*Jt*

II (c) *x* = sin*2t,* *y* = sin(t

### +

sin*2t)*

(d) x= *cos 5t,* *y* = sin*2t*

*(e) x*= *t*

### +

sin 4t,*y*=

*t2*

### +

COS*3t*sin

*2t*cos

*2t*(f)

*x*= --"4

### +

*t-*

*= 4*

_{y}### +

*t2*

II III

*y* *y* *y*

III

*y* *x*

I

*x*

IV V VI

**25-27** Use the graphs of*x* =*f(t)* and *y* =*g(t) to sketch the*
parametric curve x =*f(t),* *y = g(t).* Indicate with arrows the
direction inwhich the curve is traced as*t*increases.

### ffi

**29. Graph the curve**

*x*=

*y*

*-*

*3y3*

### +

*y5.*

### ffi

**Graph the curves**

^{30.}*y*

### =

*x5*and

*x*

### =

*y(y*

*-*1)2and find their points of intersection correct to one decimal place.

where

### °~

*1, describe the line segment that joins the points*

^{t}^{~}*P1(XI,*Yl) and

*P2(X2,*

*Y2).*

(b) Find parametric equations to represent the line segment from (-2,7) to (3, -1).

### ffi

32. Use a graphing device and the result of Exercise 31(a) to draw the triangle with*vertices A(l,*

*1), B(4, 2), and C(l, 5).*

### ITIJ

Find parametric equations for the path of a particle that moves along the circle x*2*

### +

*(y ~ 1)2 = 4 in the manner*described.

(a) Once around clockwise, starting at (2, 1)

(b) Three times around counterclockwise, starting at (2, 1) (c) Halfway around counterclockwise, starting at (0, 3)

### ffi [B:]

(a) Find parametric equations for the ellipse*x2/a2*

### +

*y2/b2*

^{=}1.

*[Hint: Modify the equations of*the circle in Example 2.]

(b) Use these parametric equations to graph the ellipse when
*a* =*3 and b* = 1, 2,4, and 8.

(c) How does the shape *of the ellipse change as b varies?*

### ffi

**Use a graphing calculator or computer to reproduce the picture.**

^{35-36}**37-38** Compare the curves represented by the parametric equa-
tions. How do they differ?

*37. (a) x* =*t**3,* *Y*=*t**2* *(b) x* =*t**6,* *Y*=*t**4*

(c) *X* =*e-**3',* *y* =*e-**2'*

38. (a) *x* ^{=}*t,* *Y* =*t-** ^{2}*
(c)

*x*

^{=}

*e',*

*y*

^{=}

*e-*

*2'*

Let *P*be a point at a distance *d*from the center of a circle of
radius *r.*The curve traced out by P as the circle rolls along a
straight line iscalled a trochoid. (Think of the motion of a
point on a spoke of abicycle wheel.) The cycloid is the spe-
cial case of a trochoid with *d*=*r. Using the same parameter*
*(J as for the cycloid and, assuming the line is the x-axis and*
*(J*=

### °

^{when}

^{P}^{is at}

^{one}of its lowest points, show that parametric equations of the trochoid are

### 11[]

*If a and b are fixed numbers, find parametric*equations for the curve that consists of all possible positions of the point

*P*

*in the figure, using the angle (J as the parameter. Then*elimi- nate the parameter and identify the curve.

*42. If a and b are fixed numbers,* find parametric equations for
the curve that consists ofall possible positions of the point *P*
in the figure, *using the angle (J as the parameter. The line seg-*
ment *AS* is tangent to the larger circle.

A curve, called a witch of Maria Agnesi, consists ofall pos-
sible positions of the point *P* in the figure. Show that para-
metric equations for this curve can be written as

*x = 2a cot (J*
Sketch the curve.

(a) Find parametric equations for the set of all points *P* as
shown in the figure such that lOP 1= 1*AB*I. (This curve
is called the cissoid of Diodes after the Greek scholar
Diocles, who introduced the cissoid as a graphical method
for constructing the edge of a cube whose volume is twice
that of a given cube.)

(b) Use the geometric description ofthe curve to draw a rough sketch of the curve by hand. Check your work by using the parametric equations to graph the curve.

(a) Graph the paths of both particles. How many points of intersection are there?

(b) Are any of these points of intersection *collision* *points?*

In other words, are theparticles ever at the same place at the same time? Ifso,find the collision points.

(c) Describe what happens if the path of the second particle is given by

If a projectile isfiredwith an initial velocity of*Vo* meters per
second at an angle 0'above the horizontal and air resistance
isassumed to be negligible, then itsposition after *t seconds* is

### l/d~ORATORY

**PROJECT**

*(a,*0)

*A* *x*

### ilI3

LookatModule 11.1 Bto see how hypocycloids and epicycloids are formed by the motion of rolling circles.where *9*isthe acceleration due to gravity (9.8 m/s2).

(a) If a gun is fired with 0'= 30°and *Vo* = 500 mis, when
will the bullet hit the ground? How far from the gun will
it hit the ground? What isthe maximum height reached
by the bullet?

~ (b) Use agraphing device tocheck your answers to part (a).

Then graph the path ofthe projectile for several other values of the angle 0'to see where it hits the ground.

Summarize your findings.

(c) Show that the path isparabolic by eliminating the parameter.

### ~ l1LJ

Investigate the family of curves defined by theparametric equations*x*=

*t*

*2,*

*Y*=

*t*

*3*

*-*

*ct.*Howdoes the shape change as cincreases? Illustrate by graphing several members of the family.

~ 48. The swallowtail catastrophe curves are defined bythepara-
metric equations *x*

### =

*2ct - 4t*

*3,*

*Y*

### =

*-et*

*2*

### +

*3t*

*4•*Graph several of these curves. What features do the curves have in common? How do they change when

*c increases?*

~ ~ The curves with equations *x* ^{=}*a sinnt, y* ^{=}*b cas tare*
called Lissajous figures. Investigate how these curves vary
when a, b,*and n vary.* (Take *n*to be apositive integer.)

Investigate the family of curves defined bythe parametric equations x

### =

cas*t,y*

### =

sin t - sin*ct,*where e

### >

O. Start by letting*c*be a positive integer and see what happens tothe shape asc increases. Then explore some ofthe possibilities that occur when eis afraction.

~ RUNNING CIRCLES AROUND CIRCLES

In this project we investigate families of curves, *called hypocycloids* and *epicycloids,* that are
generated by the motion of a point on acircle that rolls inside oroutside another circle.

I. A hypocycloid is acurve traced out by a fixed point *P*ona circle Cof radius *b*as C rolls on
the inside ofacircle with center 0and radius *a.*Show that if theinitial position of*P*is*(a,**0)*
and the parameter

*e*

is chosen as in the figure, then parametric equations of the hypocycloid
are
*(a -* *b* *)*

*x* =*(a -* *b)cas*

*e* +

*bcas*

*-b-*

*e* *y*

=*(a-*

*b)sin*

*e* *-*

*bsin*

^{(a}-b-^{-}

^{b}*e*

^{)}2. Use a graphing device (or the interactive graphic in TEC Module 11.IB) todraw the graphs
ofhypocycloids with *a*a positive integer and b ^{=} 1.How does the value of*a*affect the graph?

Show that if we take a=4,then the parametric equations ofthe hypocycloid reduce to

about the x-axis. Therefore, from Formula 7, we get S=So" 27fTsin t

*J(* *-r*

sin*t)2*

### + *(r*

cos t)2*dt*

=27T

*fo'T* *r*

^{sin t}

*Jr*

^{2}*(sin*

*2t*

### + *cos*

*2t) dt*=27T

### So" *r*

^{sin t •}

*r*

^{dt}### o

^{EXERCISES}

1-2 Find dy/dx.

I. *x* =*t* *- t**3,* *Y* ^{=}*2 - 5r*

3-6 Find anequation ofthe tangent to the curve at the point cor- responding tothe given value of the parameter.

*3. x* = *r**2*

### +

*t,*

*Y*=

*t*

*2*

*-*

*t;*

*t*= 0

*4.*

*x*=

*t -*

*r-*

^{1,}*y*= I

### +

*t*

^{2;}*t*

^{=}### W

*x*=

*e*

*Ji,*

*y*=

*t*

*- Int*

*2;*

*t*= I

*6.*

*x*=

*t*sin t,

*y*=

*t cos*

*t;*

*t*= 7T

7-8 Find an equation ofthe tangent to the curve at the given point by two methods: (a) without eliminating theparameter and (b) by first eliminating the parameter.

*7. x* = I

### +

In*t,*

*y*=

*t*

*2*

### +

2; (1,3)*8. x*= tan

*e,*

*y*= sec e;

*(I,-li)*

~ 9-10 Find anequation of the tangent(s) tothe curve at the given point. Then graph the curve and the tangent(s).

*9. x* ^{=}6 sin t, *y* ^{=}*t**2*

### +

*t;*(0,0)

*10.x=cost+cos2r,* *y=sint+sin2t;* *(-1,1)*

11-16 Find *dy/dx* *and d**2**y/dx**2•* For which values *of t is*the
curve concave upward?

*12.* *x=t**3**-12t,* *y=t**2-*

*14.* *x* *= t*

### +

In*t,*

*y*=

*t -*In

*t*

*15.*

*x*=2 sin

*t,*

*y*=3 cos

*t,*0

### <

*t*

### <

^{27T}

*16.x* =cos*2r,* *y* =cos*r,* 0<*r*<7T

17-20 Find the points on the curve where the tangent is horizon- tal orvertical. If you have a graphing device, graph the curve to check your work.

*17.* *x* = 10 - *t**2,* *Y*=*r**3* *-* *12t*

*18.* *x* =*2r**3*

### +

*3t*

*2*

*-*

*12t,*

*Y*=

*2t*

*3*

### +

*3t*

*2*

*+*

*19.* *x* =2 cos*e,* *y* =sin*2e*
*20.* *x* =cos *3e,* *y* =2sin*e*

~ 21. Use a graph toestimate the coordinates of the rightmost point
on the curve *x*=*t* *-* *t**6,* *Y*=*e'.*Then use calculus tofind the
exact coordinates.

~ 22. Use a graph to estimate the coordinates of the lowest point
and the leftmost point on the curve *x* ^{=}*t**4* *-* *2t, y* ^{=}*r*

### +

*t*

*4•*

Then findthe exact coordinates.

~ 23-24 Graph the curve in a viewing rectangle that displays all the important aspects of the curve.

### InJ

*x*=

*r*

*4*

*-*

*2t*

*3*

*-*

*2t*

*2,*

*Y*=

*t*

*3*

*-*

*t*

*24.*

*x*

^{=}

*t*

*4*

### +

*4r*

*3*

*-*

*8t*

*2,*

*y*

^{=}

*2t*

*2*

*-*

*t*

~ Show that the curve x ^{=}cos t,*Y*^{=}sin *t* cos*t has* two
tangents at (0,0) and find their equations. Sketch the curve.

~ 26. Graph the curve *x* ^{=}cos t

### +

2 cos*2t, y*

^{=}sin t

### +

2sin*2r to*discover where itcrosses itself. Then find equations of both tangents at that point.

27. (a) Find the slope of the tangent line tothe trochoid
*x* =

*re* *-*

^{d}^{sin}

*e,* *y*

^{=}

*r -*

^{d}^{cos}

*e*

in terms of *e.*

^{(See}

Exercise 40 in Section 11.1.)

(b) Show that if*d*

### <

*r,*then thetrochoid does not have a vertical tangent.

28. (a) Find the slope of the tangent tothe astroid *x* =*a*cos3

### e,

*y* =*a*sin3

### e

interms of*e.*(Astroids are explored in the Laboratory Project on page 665.)

(b) Atwhat points isthetangent horizontal orvertical?

(c) At what points does the tangent have slope I or - I?

29. At what points on the curve x =*2r**3,* *y* =I

### +

*4r*

*-*

*t*

*2*does the tangent line have slope I?

30. Find equations of thetangents to the curve x *= 3r**2*

### +

I,*Y*=

*2t*

*3*

### +

1 that pass through the point (4,3).### 1m

^{Use}

^{the}

^{parametric}

^{equations}

^{of}

^{an}

^{ellipse, x}

^{=}

^{a cos}*e,*

*y*

=*b*sin

*e,*

0~ *e* *~*

27T, tofind the area that itencloses.
32. Find the area enclosed by the curve *x*

### =

^{t}

^{2}

^{-}

^{2t,}*Y*

### = *Jt*

^{and}

they-axis.

33. Find the area enclosed by thex-axis and the curve
*x* = 1

### +

*e', y*=

*t -*

*t*

*2•*

34. Find the area of the region enclosed by the astroid

*x* = *a cos*3(), *y* =*a sin*3(). (Astroids areexplored in the Labo-
ratory Project on page 665.)

*y*

35. Find the area under one arch of the trochoid of Exercise 40 in Section 11.1 for the case d

### <

*r.*

36. Let m be the region enclosed by the loop of the curve in Example 1.

(a) Find the area ofm.

(b) Ifm isrotated about the x-axis, find the volume of the resulting solid.

(c) Find the centroid ofm.

**37-40** Set up an integral that represents the length ofthe curve.

Then use your calculator to find the length correct to four decimal places.

*37.* *x* = *t* *-* *t\* *y* = *~t3/\* I",; *t**",;*2
*38.* *x* ^{=} 1

### +

*e',*

*y*

^{=}

*t*

*2,*-3"';

*t ",;*3

*39.*

*x*=

*t*

### +

cos*t,*

*Y*=

*t*

*-*sin

*t,*0",;

*t ",;*

*27T*

*40.*

*x*

*= In t,*

*y*=

### Jt+l,

^{1",;}

^{t}^{",;}^{5}

**41-44** Find the exact length of the curve.

### IiQ

*x*= 1

### +

*3t*

*2,*

*Y*=4

### +

*2t*

*3,*0",;

*t ",;*

*42.* *x* = *e'*

### +

*e-',*

*y*= 5 -

*2t,*0",;

*t*

*",;*3

*43.*

*x*= --

*t*

*y*=In(1

### +

*t),*0",;

*t",;*2

I

### +

*t'*

**~** **45-47** Graph the curve and find itslength.

*~* *x* =*e'*cos*t,* *y* =*e' sin t,* 0",; *t ",;**7T*

*46.* *x* =cos*t*

### +

In(tan*~t),*

*y*= sin

*t,*

*7T/4"';*

*t*

*",; 37T/4*

*47.*

*x*=

*e'*

*-*

*t,*

*Y*=

*4e,/2,*-8"';

*t",;*3

48. Find the length of the loop of the curve *x* =*3t* *-* *t3,*
*Y* *= 3t**2•*

49. Use Simpson's Rule *with n* =6toestimate the length ofthe
curve x =*t -* *e', y* =*t*

### +

*e',*-6"';

*t ",;*6.

50. InExercise 43 in Section I 1.1you were asked toderive the
parametric equations x =*2a cot (), y* =*2a* sin2() for the
curve called the witch of Maria Agnesi. Use Simpson's Rule
*with n* ^{=}4 to estimate the length of the arc ofthis curve
given by *7T/4 ",; ()**",;**7T/2.*

**51-52** Find the distance traveled bya particle with position *(x, y)*
as*t*varies in the given time interval. Compare with the length of
the curve.

**51.** *x*

### =

*sin*

*2*

*t,*

*y*

### =

*cos*

*2*

*t,*0",;

*t",;*

*37T*

*52.*

*x*

### =

*cos*

*2*

*t,*

*Y*

### =

cos*t,*0",;

*t ",;*

*47T*

53. Show that the total length ofthe ellipse *x* =*a* sin (),
*y* =*b* cos(),a

### >

*b*

### >

0,is*L*=*4a**So"/2 JI* - *e**2 sin*2() *d()*

where *e*isthe eccentricity ofthe ellipse *(e* =c /*a,*where
c=

*Ja*

^{2}

^{-}

^{b}

^{2}

^{).}54. Find the total length of the astroid *x* =*a*cos3(), *y* =*a sin*3(),

where *a>* O.

### [ill]

55. (a) Graph the epitrochoid with equations*x* ^{=} *II cos t* *-* 4 cos(llt/2)
*Y*= II sin *t* *-* 4*sin(llt/2)*

What parameter interval gives the complete curve?

(b) Use your CAS tofind the approximate length ofthis curve.

### [ill]

56. Acurve called Cornu's spiral isdefined by the parametric equations*x* = *C(t)* =

*f~*

^{COS(7Tu2}

^{/2)}

^{du}*y* = *S(t)* =

### t

^{sin(7Tu}

^{2}

^{/2) du}

where CandSare the Fresnel functions that were introduced inChapter 5.

(a) Graph this curve. What happens as*t~* 00and as
*t ~* -oo?

(b) Find the length ofCornu's spiral from the origin tothe point with parameter value t.

**57-58** Set upan integral that represents the area ofthe surface
obtained by rotating the given curve about the *x-axis.* Then use
your calculator tofind the surface area correct tofour decimal
places.

*57.* *x* = 1

### +

*tel,*

*y*=

*(t2*

### +

*I)e',*0",;

*t",;*I

*58.*

*x*=

*sin*

*2*

*t,*

*y*

^{=}sin 3t, 0",;

*t ",; 7T/3*

**59-61** Find the exact area of the surface obtained by rotating the
given curve about the x-axis.

*59. x* = *t**3,* *Y*= *t**2,* 0 ~ *t ~ I*

*60. x* = *3t - t',* *y* = *3t**2,* *0 ~ t ~ I*

*~* *x* ^{=}*a cos*3

### e, *y*

^{=}

*a sin*3

### e,

0 ~*e ~*

7[/2
### ffi

62. Graph the curve*x* ^{=}2 cos

*e -*

cos*2e*

If this curve isrotated about the x-axis, find the area ofthe resulting surface. (Use your graph to help find the correct parameter interva1.)

63. Ifthe curve

*y=t--* I
*t**2*

is rotated about the x-axis, use your calculator to estimate the area of the resulting surface to three decimal places.

64. If the arc of the curve in Exercise 50 is rotated about the x-axis, estimate the area of the resulting surface using Simp- son's Rule with n=4.

**65-66** Find the surface area generated by rotating the given
curve about the y-axis.

*~* *x* = *3t**2,* *Y*=*2t**3,* 0 *~ t ~ 5*
*66. x = e' - t,* *Y =4e'/2,* *0 ~ t ~ I*

67. If

*l'*

is continuous and1'(t) *~ 0 for a ~ t ~ b,*show that the parametric curve x =

*f(t),*

*y = g(t), a ~ t~*

*b, can be put in*

*the form y*=

*F(x).*

*[Hint: Show that f-I*exists.]

68. Use Formula 2 to derive Formula 7 from Formula 9.2.5 for
the case inwhich the curve can be represented in the form
*y* ^{=}*F(x), a ~ x ~* *b.*

69. The curvature at a point P of a curve is defined as

I

### d¢1

*K-* *-*

*ds*

where ¢is the angle of inclination of the tangent line at*P,*
as shown in the figure. Thus the curvature is the absolute
value of the rate of change of ¢with respect to arc length. It
can be regarded as a measure of the rate of change of direc-
tion of the curve at*P*and will be studied in greater detail in
Chapter 14.

(a) For aparametric curve x = x(t), Y = y(t), derive the formula

where the dots indicate derivatives with respect to t, so

*x*

^{=}

^{dx/dt.}

^{[Hint:}^{Use}

^{¢}

^{=}

*tan-I(dy/dx)*and Formula 2

*to find d¢/dt.*Then use the Chain Rule

*to find d¢/ds.]*

(b) *By regarding a curve y* =*f(x)* as the parametric curve
*x* =*x, y* =*f(x),* *with parameter x,* show that the formula
in part (a) becomes

70. (a) Use the formula in Exercise 69(b) to find the curvature of
*the parabola y* ^{=}*x**2* at the point (I, I).

(b) At what point does this parabola have maximum curvature?

**71.** Use the formula in Exercise 69(a) to find the curvature ofthe
*cycloid x =*

*e -*

sin*e,* *y*

= I - cos*e*

at the top of one ofits
arches.
72. (a) Show that the curvature at each point of a straight line
is *K* =O.

(b) Show that the curvature at each point of a circle of
*radius r is* *K* = *l/r.*

73. A string is wound around a circle and then unwound while
being held taut. The curve traced by the point *P*at the end of
the string is called the involute of the circle. If the circle has
*radius r and center 0 and the initial position of* *P**is (r,* *0),*
and if the parameter

*e*

is chosen as in the figure, show
that parametric equations of the involute are
74. *A cow is tied to a silo with radius r by a rope just long*
enough to reach the opposite side of the silo. Find the area
available for grazing bythe cow.

### o

^{EXERCISES}

**1-2** Plot the point whose polar coordinates are given. Then find

two other pairs ofpolar coordinates ofthis point, one with *r*

### >

0andone with *r*

### <

O.**FIGURE** **18**

*r*=sin(88/5)

• In Exercise 55 you areasked to proveanalyti- cally what we have discovered from the graphs in Figure19.

**FIGURE** **19**

Members ofthe family of
lima90ns *r*=1

### +

csin 8I. (a) (2,7f/3) 2. (a) (1,77f/4)

(b) (I, -37f/4)

(b) (-3,7f/6)

and so we require that

*16n7T/5*

be an even multiple of 7T.This will first occur when
*n*=5.Therefore we will graph the entire curve if we specify that 0 <:;:

*e*

^{<:;:}

^{107T.}

Switching from

*e*

^{to t,}^{we}have the equations

**i.!J**

^{EXAMPLE}**Investigate the family of polar curves given by**

^{II}*r*

=1 ### +

c sin*e.*

How
does the shape change as c changes? (These curves are called limac;ons, after a French
word forsnail, because of the shape of thecurves for certain values of*c.)*

SOLUTION Figure 19 shows computer-drawn graphs for various values ofc. For c

### >

I there is aloop that decreases insize ascdecreases. When c= 1 the loop disappears and the curve becomes thecardioid that we sketched in Example 7. For cbetween 1and ~the cardioid's cusp is smoothed out and becomes a"dimple." When cdecreases from ~ to 0, the lima~on is shaped like an oval. This oval becomes more circular asc ~ 0, and when c### =

0 the curve isjust the circle*r* =

1.
The remaining parts of Figure 19 show that as cbecomes negative, the shapes change in reverse order. Infact, these curves arereflections about thehorizontal axis of the corre-

sponding curves with positive c. 0

(c) (-1, 7f/2) (c) (I,-I)

**5-6** The Cartesian coordinates of apoint are given.

(i) Find polar coordinates *(r,* 8)ofthepoint, where r>0 and
0,,:; 8

### <

27f.(ii) Find polar coordinates *(r,*8)ofthe point, where *r*

### <

0and 0,,:; 8<27f.**3-4** Plot the point whose polar coordinates are given. Then find

the Cartesian coordinates of the point. 5. (a) (2,-2)

6. (a)

### (3.)3, 3)

(b)

### (-1,.)3)

(b) (I, -2)

7-12 Sketch the region in the plane consisting of points whose polar coordinates satisfy the given conditions.

7. I "S *r*"S 2

*8.* *r ~* 0, 7r/3 "S

*e*

^{"S}

^{27r/3}

9. O"S *r*

### <

4, -7r/2"S*e* <

7r/6
10. 2 ### <

*r"S*5, 37r/4

### < *e* ^{<}

^{57r/4}

### [II;]

^{2}

^{<}

^{r}^{<}

^{3,}57r/3 "S

*e*

"S 77r/3
13. Find the distance between the points with polar coordinates (2, 7r/3) and (4, 27r/3).

14. Find a formula for the distance between the points with polar
coordinates *(rl,* *(1) and* *(r2,* *(2).*

15-20 Identify the curve byfinding a Cartesian equation for the curve.

*15. r*=2

### [ill

^{r}^{=}

^{3 sin}

*e*

*16.* *r cos*

*e*

^{=}

^{I}

*18. r*=2 sin

*e* +

2 cos*e*

*20.* *r*=tan

*e*

^{sec}

*e*

21-26 Find a polar equation for the curve represented by the given Cartesian equation.

21. *Y* ^{=}5
*23.x=_y2*

*~* *x**2*

### +

^{y2}*= 2ex*

*24. Y*=*2x* *-* 1
*26. xy* =4

27-28 For each of the described curves, decide if the curve would be more easily given by a polar equation or a Cartesian equation.

Then write an equation for the curve.

27. (a) A line through the origin that makes an angle of 7r/6 with the positive x-axis

(b) A vertical line through the point (3,3) 28. (a) A circle with radius 5and center (2,3)

(b) A circle centered atthe origin with radius 4

29-48 Sketch the curve with the given polar equation.

*29.*

*e*

^{=}

^{-7r/6}

^{30. r}

^{2}

^{-}

^{3r}^{+}

^{2}

^{=}

^{0}

31. *r* =sin

*e*

*33.* *r* =2(1 - sin

*e), e ~*

0
*~r=*

*e, e~o*

*37.* *r = sin**2e*

### m

^{r}^{=}

^{2cos4e}*41.* *r = 1*- 2 sin

*e*

32. *r*= *-3 cos e*
34. *r*= *1 - 3 cos e*
36. *r*= *In e, e~1*
38. *r*= 2 cos*3e*
40. *r*= *3 cos 6e*
42. *r*= 2

### +

*sin e*

*43.* *r**2* = 9 sin*2e*
*45. r*=2*cos(3e/2)*
*47.* *r* =I

### +

2 cos*2e*

*44.* *r**2* =cos *4e*
*46.* *r**2*

*e*

^{= I}

*48.* *r*= 1

### +

*2 cos(e/2)*

49-50 The figure shows the graph of*r*as a function of

*e*

^{in Carte-}

sian coordinates. Use it to sketch the corresponding polar curve.

51. Show that the polar curve *r* =4

### +

2 sec*e*

(called a conchoid)
has the *line x*=2 asavertical asymptote by showing that lim,~"xx

^{=}2. Use this fact to help sketch the conchoid.

52. Show that the curve r= 2 - csc

*e*

(also a conchoid) has the
line *y*= -1 as a horizontal asymptote by showing that lim,~"x

*y*=-I. Use this fact to help sketch the conchoid.

53. Show that the curve r =sin

*e*

^{tan}

*e*

(called a cissoid of
Diodes) *has the line x*= I as a vertical asymptote. Show also that the curve liesentirely within the vertical strip 0 "S

*x*

### <

I.Use these facts to help sketch the cissoid.

54. *Sketch the curve (x**2*

### +

y2)J*= 4X*

*2*

*y*

*2.*

~ (a) In Example 11the graphs suggest that the lima<;on
*r* ^{=} I

### +

e sin*e*

has an inner loop when ### I

e### I >

I. Prove that thisistrue, and find the values of*e*

that correspond to
the inner loop.
(b) From Figure 19 it appears that the lima<;on loses its dimple when e =~.Prove this.

56. Match the polar equations with the graphs labeled I-VI. Give
reasons for your choices. (Don't use a graphing device.)
*(a) r*

### =)8,

*0 "S e "S 167r*

*(b) r*

### =

*e*

*2,*

*0 "S e*"S167r

(c) *r* =*cos(e/3)* (d) *r*= I

### +

*2cos e*

*(e) r*=2

### +

*sin 3e*(f)

*r*= 1

### +

*2 sin 3e*

I II

57-62 Find the slope ofthetangent line to the given polar curve atthe point specified by the value of O.

*57.* *r* =2 sinO, *0*=17/6

### rru

^{r = I/O,}

^{0}^{=}

^{17}

*61.* *r*^{=}cos 20, *0*^{=}17/4

*58.* *r =*2 - sin 0, *0*= 17/3
*60.* *r*=*cos(0/3),* *0*= 17

### ffi

^{80. A}family of curves is given bythe equations

*r*= I

### +

*c sin nO,*where

*c is a real number and n is a positive*integer. How does the graph change as

*n*increases? How does it change as

*c*changes? Illustrate by graphing enough members of the fam- ilyto support your conclusions.

63-68 Find the points on the given curve where the tangent line ishorizontal or vertical.

~ Show that the polar equation *r*=*a sin 0*

### +

*b cos0,*where

*ab*# 0,represents a circle, and find its center and radius.

70. Show that the curves *r*=*a*sin*0 and* *r*=*a cos0 intersect at*
right angles.

### ffi

^{71-76}

^{Use}

^{a}graphing device to graph the polar curve. Choose the parameter interval tomake sure that you produce the entire curve.

*71.* *r*= I

### +

2sin(0/2) (nephroid of Freeth)*72.*

*r*= JI - 0.8 sin20 (hippopede)

*73. r*

*= es;no*

*-*

*2 cos(40)*(butterfly curve)

*74. r*=sin2(40)

### +

*cos(40)*

*75.* *r* = 2 - 5sin(0/6)
*76. r*=*cos(0/2)*

### +

*cos(0/3)*

### ffi

^{77.}How are the graphs of

*r*= I

### +

sin(O - 17/6) and*r*^{=} I

### +

sin(O - 17/3)related tothegraph*of r*

^{=}I

### +

sin O?Ingeneral, how isthe graph *of r* ^{=}*f(O* *-* *a)*related tothe
graph *of r*=*f(O)?*

### ffi

^{78. Use a}

^{graph to}

^{estimate}

^{the}y-coordinate of the highest points on the curve

*r*=sin 20.Then use calculus tofind the exact value.

### ffi

^{79.}(a) Investigate the family of curves defined bythe polar equa-

tions *r* =sin nO,where n is apositive integer. How isthe

number of loops related to n?

(b) What happens ifthe equation in part (a)is replaced by
*r=* IsinnOI?

*-* *a cos 0*
*r=*

*+*

*a cos0*

Investigate how thegraph changes asthe number *a*changes.

In particular, you should identify the transitional values of*a*
for which the basic shape ofthe curve changes.

### ffi

82. The astronomer Giovanni Cassini (1625-1712) studied the family ofcurves with polar equationswhere a and *c are* positive real numbers. These curves are
called the ovals of Cassini even though they are oval shaped
only for certain values of*a*and*c.*(Cassini thought that these
curves might represent planetary orbits better than Kepler's
elli pses.) Investigate the variety ofshapes that these curves
may have. In particular, how are aand *c*related to each other
when the curve splits into two parts?

83. Let P be any point (except the origin) onthe curve *r* =*f(O).*

If

### l/J

isthe angle between the tangent line at Pand the radial line OF,show thattan l/J= -/-*r*
*dr* *dO*

84. (a) UseExercise 83to show that the angle between the tan- gent line and the radial line is

### l/J

^{=}17/4 at every point on the curve

*r =*

*eO.*

### ffi

(b) Illustrate part (a) bygraphing the curve and the tangent lines at the points where*0*

^{=}0and 17/2.

(c) Prove that any polar curve *r* =*f(O)* with the property that
the angle l/J between the radial line and the tangent line is
a constant must be ofthe form *r*=*Ce**kO,* where Cand *k*
are constants.

FIGURE 8
*r*=I

### +

sin 0Formula 5gives

*L*

### = *J:7T* *)* *r* *2* + (* r ^{de} ^{=} ^{fo} ^{27T}

^{de}

^{fo}

^{27T}

^{)(1}

^{+}

^{sin 8)2}

^{+}

^{cos}

^{2e de}

^{2e de}

*=*

*f027T*

)2 ### +

2sin*e de*

We could evaluate this integral by multiplying and dividing the integrand by

)2 - 2 sin

*e ,*

or we could use acomputer algebra system. In anyevent, we find that the
length of the cardioid is*L* =8. 0

### -EJ

^{EXERCISES}

1-4 Find the area of the region that isbounded bythe given curve and liesin the specified sector.

5-8 Find the area ofthe shaded region.

5. 6.

17-21 Find the area of the region enclosed by one loop of the curve.

*19.* *r=3cos5e* 20. r=2sin68

### 1m

^{r}^{=}

^{I}

### +

2sin*e*

(inner loop)
22. Find the area enclosed by the loop ofthe strophoid
*r* =2cos

*e* *-*

^{sec}

*e.*

23-28 Find the area of the region that liesinside the firstcurve and outside the second curve.

*23.* *r* =2 cos*0,* *r* = I
*25.* *r* =4sin 0, *r* =2

*~* *r*

### =

3 cos*0,*

*r*

### =

1### +

cos*0*

*28.*

*r*=3sin

*e,*

^{r}^{=}

^{2}

^{-}

^{sin}

^{0}*24.* *r*= I - sin

*e,*

^{r}^{=}

^{1}

*26.* *r* =3 cos

*e,*

^{r}^{=}

^{2}

^{-}

^{cos}

^{0}29-34 Find the area ofthe region that lies inside both curves.

*29.* *r*=

### J3

^{cos}

*e,*

^{r}^{=}

^{sin}

*e*

*30.* *r*= 1

### +

cos*e,*

*= I - cos*

^{r}*e* lID

*r*

### =

sin*20,*

*r*

### =

cos*20*

*32.* *r*

### =

3### +

2 cos*0,*

*r*

### =

3### +

2sin 0*33.*

*r*

*2*=

*sin 2e,*

*r*

*2*=cos

*20*

*34. r*=*a*sin 0, *r*=*b cos0,* *a*

### >

0,*b*

### >

035. Find the area inside the larger loop and outside the smaller loop
of the limac;on *r* =~

### +

cos*e.*

36. Find the area between alarge loop and the enclosed small loop
of the curve *r* = I

### +

2 cos*3e.*

37-42 Find allpoints of intersection ofthe given curves.

*37.* *r* = I

### +

sin*e,*

^{r}^{=}

^{3}

^{sin}

*e*

*38. r*= 1 - cos 0, *r* = I

### +

sin*e*

*39.* *r*

### =

2 sin*2e,*

*r*

### =

1### [ill

*r*= sin e,

*r*= sin

*2e*

*40.* *r*^{=}cos*30,* *r*^{=}sin*30*
*42. r**2*

### =

sin 2e,*r*

*2*

### =

cos*2e*

### rn

^{43.}

^{The}points of intersection of the cardioid

*r*= I

### +

sin*0 and*the spiral loop r = 20, -17/2 ""

*0""*17/2, can't befound exactly. Use a graphing device to find the approximate values of

*0 at which they intersect.*Then use these values to estimate the area that lies inside both curves.

44. When recording live performances, sound engineers often use amicrophone with a cardioid pickup pattern because itsup- presses noise from the audience. Suppose the microphone is placed 4 m from the front ofthe stage (as in the figure) and the boundary ofthe optimal pickup region is given by the car- dioid r =8

### +

8 sin*0,*where r ismeasured in meters and the microphone isat the pole. The musicians want to know the area they will have onstage within the optimal pickup range ofthe microphone. Answer their question.

**45-48** Find the exact length of the polar curve.

45. r^{=}3 sin*0,* 0 ""*0""* 17/3 *46. r*^{=}*e**20,* 0 "" 0 ""217

**49-52** Use acalculator to find the length of the curve correct to
four decimal places.

### rn

**Graph the curve and find itslength.**

^{53-54}53. r

### =

*cos*

*4*

*(O/4)*54. r

### =

cos2(O/2)55. (a) Use Formula 11.2.7 to show that the area of the surface generated by rotating the polar curve

(where

*f'*

is continuous and 0 "" a ### <

*b ""*17) about the polar axis is

### s

^{=}

### r

^{217r}^{sin}

^{0 ~}^{r}

^{2}

^{+} ^{(:~)}

^{2}

^{dO}(b) Use theformula in part (a) to find the surface area gener-
ated byrotating the lemniscate *r**2* =cos 20 about the
polar axis.

56. (a) Find a formula for the area of the surface generated by
rotating the polar curve *r* =*f(O),* *a""* *0 ""* *b* (where

*f'*

is
continuous and 0 ""*a*

### <

*b ""*17),about the line

*0*= 17/2.

(b) Find the surface area generated byrotating the lemniscate
r2 =*cos 20 about the* line 0= 17/2.

### =================~

CO N IC SECTIO NSIn this section we give geometric definitions of parabolas, ellipses, and hyperbolas and derive their standard equations. They are called conic sections, or conics, because they result from intersecting a cone with aplane as shown in Figure I.

**FIGURE** **I**
Conics

*y-l=-2:(x-4)*3
/
/
/
/
/
(4,4) //

\ /

\ / X(4,1)

FIGURE 15

*9x**2* *-* *4l-* *72x*

### +

*8y*

### +

176=0### B

^{EXERCISES}

i.'J EXAMPLE7 Sketch the conic

*9x2 - 4y**2* *-* *72x*

### +

*8y*

### +

176 =0*4(y2 - 2y) - 9(x2 -* *8x)* = 176

*4(y2 - 2y*

### +

1) -*9(x2 -*

*8x*

### +

16) = 176### +

4 - 144*4(y -*1)2 -

*9(x - 4)Z*= 36

*(y* *-* 1)2 *(x* *-* 4)2

---= 1

### 9 4

This is in the form (8) except *that x and* *y* *are replaced by x - 4 and* *y* *-* 1.Thus
*a**2* =*9, b**2* =4, and c2 = 13. The hyperbola is shifted four units to the right and one
unit upward. The foci are

### (4,

1### + /13)

and### (4,

1-### /13)

and the vertices are (4, 4) and (4, -2). The asymptotes are y - 1^{=}

*±:~(x - 4).*The hyperbola is sketched in

Figure 15. 0

1-8 Find the vertex, focus, and directrix of the parabola and sketch its graph.

I. *x =2y**2*

*3. 4x**2**=-y*

### [I]

*(x*

### +

2)2 =*8(y*

*-*3)

*7. l*

*+ 2y + 12x +*25 = 0

*2. 4y*

### +

*x*

*2*= 0

*4. y2*

*= 12x*

*6. x-I*=

*(y*

### +

5)2*8.*

*Y*

### +

*12x - 2x*2= 16

9-10 Find an equation ofthe parabola. Then find the focus and directrix.

11-16 Find the vertices and foci of the ellipse and sketch its graph.

*x2* *y2*

11.-+-=1

16 4

*x2* *y2*

12.

### -+-=

164 100

*13. 25x2*

### +

*9y*

*2*= 225 14.

*4x2*

### +

*25y2*=25

*~* *9x**2* *-* *18x*

### +

*4y*

*2*=27

*16. x**2* *+ 3l* *+ 2x -* *12y +* 10= 0

*Y*

*/V* -...

### ~

_{-1}

### 'I'--

^{2}

*V*

^{/}

*y.*

/ "-

### L ^{\}

### ~

_{1}

0 1

### 1\ ^{/}

'\ *V*

19-24 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.

*x2* *y2*

### [ill---=1

144 25

*y2* *x2*

20. - - - = 1

16 36

21. */-x2=4* *22.9x**2**-4y2=36*

*23. 4x**2* *-* *l* *-* *24x* *-* *4y*

### +

28^{=}0

*24. l*

*- 4x*

*2*

*-*

*2y*

### +

*16x*

^{=}31

25-30 Identify the type of conic section whose equation isgiven and find the vertices and foci.

*25.* *x**2* =*Y*

### +

1### ITIJ

^{x}

^{2}

^{= 4y}

^{-}

^{2y}

^{2}*29.* *y2*

### +

*2y*=

*4x*

*2*

### +

3*26. x**2* =*y2*

### +

1*28.*

*y2 -*

*8y*=

*6x*

*-*16

*30. 4x*

*2*

### +

*4x*

### +

*y2*=0

**31-48** Find anequation for theconic that satisfies thegiven
conditions.

**31.** Parabola, vertex (0,0), focus (0, -2)
**32.** Parabola, vertex (1,0), directrix *x* ^{=}^{-}5

~ Parabola, focus (-4,0), directrix x =2
**34.** Parabola, focus (3,6), vertex (3,2)
**35.** Parabola, vertex (0,0), axisthe x-axis,

passing through (I,-4)

**36.** Parabola, vertical axis, passing through (-2, 3),(0,3),
and (I,9)

### IEJ

^{Ellipse,}

^{foci}(::'::2,0), vertices (::'::5,0)

**38.**Ellipse, foci (0, ::'::5), vertices (0, ::'::13)

**39.**Ellipse, foci (0, 2), (0,6), vertices (0,0), (0,8)

**40.**Ellipse, foci (0, - I), (8, -1), vertex (9, -1)

**41.** Ellipse, center (- I,4), vertex (- I,0), focus (- I,6)
**42. Ellipse,** foci (::'::4,0), passing through (-4, 1.8)
**43.** Hyperbola, vertices (::'::3,0), foci (::'::5,0)
**44. Hyperbola,** vertices (0, ::'::2), foci (0, ::'::5)
**45.** Hyperbola, vertices (- 3, -4), (- 3,6),

foci (-3, -7), (-3,9)

**46. Hyperbola,** vertices (-1,2), (7,2),
foci (- 2,2), (8,2)

~ Hyperbola, vertices (::'::3,0), asymptotes *y* =*::'::2x*

**48.** Hyperbola, foci (2,0), (2,8),

asymptotes y = 3

### +

*~x*and

*y*= 5 -

*~x*

**49.** The point inalunar orbit nearest the surface of the moon is
called perilune and the point farthest from the surface is called
*apolune. The Apollo* *II* spacecraft was placed in anelliptical
lunar orbit with perilune altitude 110 km and apolune altitude
314 km (above the moon). Find anequation ofthis ellipse if
the radius of the moon is 1728km and the center of the moon
is atone focus.

**50.** A cross-section of aparabolic reflector is shown in the figure.

The bulb islocated atthefocus and theopening at the focus is 10 cm.

(a) Find an equation of the parabola.

(b) Find the diameter of the opening

### I

CD### I,

11 cm from the vertex.5cm
11cm-
*F*
5cm

**51.** In the LORAN (LOng RAnge Navigation) radio navigation
system, two radio stations located at*A* and *B*transmit simul-
taneous signals to aship or anaircraft *located at P. The*
onboard computer converts the time difference in receiving
these signals into adistance difference

### I

*PA*

### I - 1

*PB*

### I,

and this, according to the definition ofahyperbola, locates the ship or aircraft onone branch ofahyperbola (see the figure).Suppose that station B islocated 600 km due east of station A on acoastline. A ship received the signal from B 1200 micro- seconds (J..ls)before it received the signal from A.

(a) Assuming that radio signals travel ata speed of 300 m/J..ls, find anequation ofthe hyperbola on which the ship lies.

(b) Ifthe ship isdue north *of B,*howfar offthe coastline is
the ship?

600km transmitting stations

**52.** Use the definition of ahyperbola to derive Equation 6 for a
hyperbola with foci *(::,::c,*0)and vertices (::'::a, *0).*

**53.** Show that the function defined by the upper branch of the
hyperbola *y2/a2* *- x2*

*/b*

*2*= I isconcave upward.

**54.** Find an equation for theellipse withfoci (I,1) and (- 1, -1)
and major axis oflength 4.

**55.** Determine the type of curve represented by the equation

*x2* *y2*

-+---=

*k* *k* *-* 16

ineach of the following cases: (a)*k* > 16,(b)0

### <

*k*

### <

16, and (c)*k*<O.

(d) Show that all the curves in parts (a) and (b) have the same
foci, no matter what the value of*k is.*

**56.** (a) Show that the equation of the tangent line to the parabola
*y2* ^{=}*4px* atthe point *(xo, Yo)*can be written as

*YoY* =*2p(x*

### +

*xo)*

(b) What is thex-intercept of thistangent line? Use this fact to draw the tangent line.

**57.** Show thatthe tangent lines to the parabola x*2* =*4py* drawn
from anypoint on the directrix areperpendicular.

**58.** Show that if anellipse and ahyperbola have the same foci,
then their tangent lines at each point of intersection are
perpendicular.

**59.** Use Simpson's Rule with *n* = 10toestimate the length ofthe
ellipse x*2*

### +

*4y*

*2*=4.

**60.** The planet Pluto travels inanelliptical orbit around the sun
(at one focus). The length of the major axis is I.18 X 10^{10}km

and the length of the minor axis is 1.14 X 1010km. Use Simp-
son's Rule with *n* ^{=} 10to estimate the distance traveled by the
planet during one complete orbit around the sun.

**61.** Find the area of the region enclosed bythe hyperbola
*x*2/*a*2 -

*i/*

^{b}^{2}

^{=}

^{I}

^{and the}

^{vertical}line through afocus.

62. (a) Ifanellipse is rotated about its major axis, find the volume of the resulting solid.

(b) If it isrotated about itsminor axis, find the resulting volume.

64. Let *P(XI,* *Yl)* be apoint onthe hyperbola

*x*

^{2}*/a*

^{2}

^{-}*y*

*2*

*/b*

^{2}^{=}I with foci Fl and

*F*

*2*and let aand {3be the angles between the lines

*P F*

^{l,}*PF*

*and the hyperbola as shown in the figure. Prove that a={3.(This isthe reflection property ofthehyperbola. It shows that light aimed at a focus*

^{2}*F*

*2*of ahyperbolic mirror is reflected toward the other focus

*Fl')*

63. Let *PI(XI,* *yd* be apoint onthe ellipse

*x*

^{2}*/a*

^{2}### +

*y2/b*

*2*= I with foci Fl and

*F*

*2*and let a and {3bethe angles between the lines

*PF*

*I,*

*PF*

*2*and the ellipse as shown in the figure. Prove that a= {3.This explains how whispering galleries and lithotripsy work. Sound coming from one focus is reflected and passes through the other focus.

*[Hint: Use the formula in Problem 15*on page 202 to show that tan a =tan {3.]

*y*

In the preceding section we defined the parabola in terms of a focus and directrix, but we defined the ellipse and hyperbola interms of two foci. In this section we give a more uni- fied treatment ofallthree types ofconic sections interms of afocus and directrix. Further- more, ifwe place the focus at the origin, then a conic section has a simple polar equation, which provides a convenient description of the motion of planets, satellites, and comets.

### OJ

**Let F be a fixed point (called the**

^{THEOREM}*focus) and I be*a fixed line (called the directrix) in a plane. Let ebe a fixed positive number (called the eccentricity). The set of all points

*P*in the plane such that

*---=e*

### I?FI *IPII*

(that is, the ratio of thedistance from F to the distance *from I is*the constant *e)*
is a conic section. The conic is

(a) anellipse if

*e* <

I
(b) aparabola if e = I
(c) ahyperbola if*e* >

1
### §

^{EXERCISES}**1-8** Write a polar equation of a conic with the focus atthe origin
and the given data.

I. Hyperbola, eccentricity;1, directrix *y* =6
**2. Parabola,** directrix *x* = 4

### rn

^{Ellipse,}eccentricity

### t

directrix x = -5**4. Hyperbola,**eccentricity 2, directrix

*y*= -2

**5. Parabola,** vertex (4,*37T/2)*

**6.** Ellipse, eccentricity 0.8, vertex

### 0,

*7T/2)*

**7. Ellipse,**eccentricity

### t

directrix*r*=4 sec 8

**9-16** (a) Find theeccentricity, (b) identify the conic, (c) give an
equation of the directrix, and (d) sketch the conic.

**9. r =****10.** *r=* 12

### +

sin 8 3 - 10 cos 812 3

**II.** *r =* _{12.}*r=*

4 - sin 8 2

### +

2 cos 89 8

### Iill

*r=*

**14.**

*r=*

6

### +

2 cos8 4### +

5 sin 83 10

**15. r =****16.** *r=*

4 - 8 cos8 5 - 6 sin 8

### ffi

**(a) Find the eccentricity and directrix of the conic**

^{17.}*r*=

### 1/0 -

2 sin8) and graph the conic and its directrix.**(b) If this conic** is rotated counterclockwise about the origin
through an angle 37T/4, write the resulting equation and
graph its curve.

### ffi

**Graph the conic**

^{18.}*r*= 4/(5

### +

6 cos 8) and its directrix. Also graph the conic obtained by rotating this curve about the ori- gin through an*angle 7T/3.*

### ffi

**Graph the conics**

^{19.}*r*=

*e/O*

*- e cos 8) with e*= 0.4, 0.6, 0.8, and 1.0 on acommon screen. How does the value of e affect the shape of the curve?

### ffi

**(a) Graph the conics r =**

^{20.}*ed/(l*

### +

*e sin 8) for e*= 1 and var- ious values of

*d.*How does the value of d affect the shape of the conic?

(b) Graph these conics for d^{=} *1 and various values of e.*

How does the value *of e affect the shape of the conic?*

### In;]

Show that a conic with focus at the origin, eccentricity*e,*and directrix x =

*-d has polar equation*

*ed*

**22.** Show that a conic with focus at the origin, eccentricity *e,*and
directrix y ^{=}*d has polar equation*

*ed*

*+*

*e sin 8*

**23.** Show that a conic with focus at the origin, eccentricity *e,*and
*directrix y* ^{=} *-d has polar equation*

*ed*

**24. Show that the parabolas** *r* =

### c/O +

cos 8) and*r = d/(l*

*- cos 8) intersect at right angles.*

**25. The orbit of Mars around the sun is an ellipse with eccen-**
tricity 0.093 and semimajor axis 2.28 X 108km. Find a polar
equation for the orbit.

**26. Jupiter's orbit has eccentricity** 0.048 and the length of the
major axis is 1.56 X 10^{9}km. Find a polar equation for the
orbit.

### ITIJ

The orbit of Halley's comet, last seen in 1986 and due to return in 2062, is an ellipse with eccentricity 0.97 and one focus at the sun. The length of its major axis is 36.18 AU.[An astronomical unit (AU) is the mean distance between the
earth and the sun, about 1.5 X 10^{8}km.] Find a polar equation
for the orbit of Halley's comet. What is the maximum
distance from the comet to the sun?

**28.** The Hale-Bopp comet, discovered in 1995, has an elliptical
orbit with eccentricity 0.9951 and the length of the major axis
is 356.5 AU. Find a polar equation for the orbit of this comet.

How close to thesun does it come?

**29.** The planet Mercury travels in an elliptical orbit with eccen-
tricity 0.206. Its minimum distance from the sun is
4.6 X 107km. Find its maximum distance from the sun.

**30. The distance** from the planet Pluto to the sun is

4.43 X 109km at perihelion and 7.37 X 109km at aphelion.

Find the eccentricity of Pluto's orbit.

**31.** Using the data from Exercise 29, find the distance traveled by
the planet Mercury during one complete orbit around thesun.

(If your calculator or computer algebra system evaluates defi- nite integrals, use it.Otherwise, use Simpson's Rule.)

### =================0

^{REVI EW}

I. (a) What is aparametric curve?

(b) How do you sketch a parametric curve?

2. (a) How doyou find the slope of atangent to aparametric curve?

(b) How do you find the area under a parametric curve?

3. Write an expression for each of the following:

(a) The length of a parametric curve

(b) The area ofthe surface obtained byrotating a parametric curve about the x-axis

4. (a) Use adiagram toexplain the meaning ofthepolar coordi-
nates *(r,* *0) of a point.*

(b) Write equations that express the Cartesian coordinates
*(x, y) of a*point in terms of the polar coordinates.

(c) What equations would you use to find thepolar coordinates ofa point ifyou knew the Cartesian coordinates?

5. (a) How do you find the slope of atangent line toapolar curve?

(b) How do you find the area of a region bounded by apolar curve?

(c) How do you find the length of apolar curve?

6. (a) Give a geometric definition ofa parabola.

(b) Write anequation of aparabola with focus (0,*p)* and direc-
trix *y* = *-po* What ifthe focus is*(p,* 0) and the directrix
isx= *-p?*

7. (a) Give a definition ofanellipse in terms of foci.

(b) Write anequation forthe ellipse with foci *(±c,* 0)and
vertices (±a, *0).*

8. (a) Give adefinition of ahyperbola interms offoci.

(b) Write anequation for the hyperbola with foci *(±c,* 0) and
vertices *(±a,* *0).*

(c) Write equations for the asymptotes ofthe hyperbola in part (b).

9. (a) What isthe eccentricity of a conic section?

(b) What can you say about theeccentricity ifthe conic section isanellipse? A hyperbola? Aparabola?

(c) Write apolar equation for aconic section with eccentricity
*e*and directrix *x =* *d.* What ifthe directrix is*x* *= -d?*

*y* = *d?* *y* = *-d?*

Determine whether the statement is true or false.If itis true. explain why.

Ifit is false,explain why or give an example that disproves the statement.

I. If theparametric curve *x* = *f(t),* *y* *= g(t) satisfies* *g'(I)* = 0,
then it has ahorizontal tangent when *t* = I.

2. If x = *f(t)* *and y* = *get) are twice* differentiable, then

*d**2**y* *d**2**y/dt**2*

*dx**2* *d2x/dt2*

3. The length ofthecurve *x* =*f(t),* *y* *= g(t),* *a ~* *t* *~ b,*is

*.I:J[f'(t)]2*

### +

*[g'(t)]2*

*dt.*

4. If apoint isrepresented by *(x,**y)* in Cartesian coordinates
(where *x* ¥- 0) and *(r,* *0) in polar* coordinates, then

*0=* tan*-I(y/X).*

5. The polar curves *r*= 1 - sin28 and *r* = sin28 - ] havethe
same graph.

6. The equations *r* = 2, x2

### +

*y2*= 4,and x = 2 sin

*3t,*

*y*= 2 cos 3t (0 ~

*t~ 27T)*all have the same graph.

7. The parametric equations *x* = *t**2,* *Y* *= t**4* have the same graph
as x ^{=}*t**3,* *Y* =*t**6•*