<< Chapter < Page Chapter >> Page >

For the following exercises, convert the polar equation of a conic section to a rectangular equation.

r = 4 1 + 3   sin   θ

Got questions? Get instant answers now!

r = 2 5 3   sin   θ

25 x 2 + 16 y 2 12 y 4 = 0

Got questions? Get instant answers now!

r = 8 3 2   cos   θ

Got questions? Get instant answers now!

r = 3 2 + 5   cos   θ

21 x 2 4 y 2 30 x + 9 = 0

Got questions? Get instant answers now!

r = 4 2 + 2   sin   θ

Got questions? Get instant answers now!

r = 3 8 8   cos   θ

64 y 2 = 48 x + 9

Got questions? Get instant answers now!

r = 2 6 + 7   cos   θ

Got questions? Get instant answers now!

r = 5 5 11   sin   θ

96 y 2 25 x 2 + 110 y + 25 = 0

Got questions? Get instant answers now!

r ( 5 + 2   cos   θ ) = 6

Got questions? Get instant answers now!

r ( 2 cos   θ ) = 1

3 x 2 + 4 y 2 2 x 1 = 0

Got questions? Get instant answers now!

r ( 2.5 2.5   sin   θ ) = 5

Got questions? Get instant answers now!

r = 6 sec   θ 2 + 3   sec   θ

5 x 2 + 9 y 2 24 x 36 = 0

Got questions? Get instant answers now!

r = 6 csc   θ 3 + 2   csc   θ

Got questions? Get instant answers now!

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.

r = 2 3 + 3   sin   θ

Got questions? Get instant answers now!

r = 10 5 4   sin   θ

Got questions? Get instant answers now!

r = 3 1 + 2   cos   θ

Got questions? Get instant answers now!

r = 8 4 5   cos   θ

Got questions? Get instant answers now!

r = 3 4 4   cos   θ

Got questions? Get instant answers now!

r = 6 3 + 2   sin   θ

Got questions? Get instant answers now!

r ( 3 4 sin   θ ) = 9

Got questions? Get instant answers now!

r ( 3 2 sin   θ ) = 6

Got questions? Get instant answers now!

r ( 6 4 cos   θ ) = 5

Got questions? Get instant answers now!

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.

Directrix: x = 4 ; e = 1 5

r = 4 5 + cos θ

Got questions? Get instant answers now!

Directrix: x = 4 ; e = 5

Got questions? Get instant answers now!

Directrix: y = 2 ; e = 2

r = 4 1 + 2 sin θ

Got questions? Get instant answers now!

Directrix: y = 2 ; e = 1 2

Got questions? Get instant answers now!

Directrix: x = 1 ; e = 1

r = 1 1 + cos θ

Got questions? Get instant answers now!

Directrix: x = 1 ; e = 1

Got questions? Get instant answers now!

Directrix: x = 1 4 ; e = 7 2

r = 7 8 28 cos θ

Got questions? Get instant answers now!

Directrix: y = 2 5 ; e = 7 2

Got questions? Get instant answers now!

Directrix: y = 4 ; e = 3 2

r = 12 2 + 3 sin θ

Got questions? Get instant answers now!

Directrix: x = −2 ; e = 8 3

Got questions? Get instant answers now!

Directrix: x = −5 ; e = 3 4

r = 15 4 3 cos θ

Got questions? Get instant answers now!

Directrix: y = 2 ; e = 2.5

Got questions? Get instant answers now!

Directrix: x = −3 ; e = 1 3

r = 3 3 3 cos θ

Got questions? Get instant answers now!

Extensions

Recall from Rotation of Axes that equations of conics with an x y term have rotated graphs. For the following exercises, express each equation in polar form with r as a function of θ .

x 2 + x y + y 2 = 4

r = ± 2 1 + sin θ cos θ

Got questions? Get instant answers now!

2 x 2 + 4 x y + 2 y 2 = 9

Got questions? Get instant answers now!

16 x 2 + 24 x y + 9 y 2 = 4

r = ± 2 4 cos θ + 3 sin θ

Got questions? Get instant answers now!

Chapter review exercises

The Ellipse

For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.

x 2 25 + y 2 64 = 1

x 2 5 2 + y 2 8 2 = 1 ; center: ( 0 , 0 ) ; vertices: ( 5 , 0 ) , ( −5 , 0 ) , ( 0 , 8 ) , ( 0 , 8 ) ; foci: ( 0 , 39 ) , ( 0 , 39 )

Got questions? Get instant answers now!

( x 2 ) 2 100 + ( y + 3 ) 2 36 = 1

Got questions? Get instant answers now!

9 x 2 + y 2 + 54 x 4 y + 76 = 0

( x + 3 ) 2 1 2 + ( y 2 ) 2 3 2 = 1 ( 3 , 2 ) ; ( 2 , 2 ) , ( 4 , 2 ) , ( 3 , 5 ) , ( 3 , 1 ) ; ( 3 , 2 + 2 2 ) , ( 3 , 2 2 2 )

Got questions? Get instant answers now!

9 x 2 + 36 y 2 36 x + 72 y + 36 = 0

Got questions? Get instant answers now!

For the following exercises, graph the ellipse, noting center, vertices, and foci.

x 2 36 + y 2 9 = 1

center: ( 0 , 0 ) ; vertices: ( 6 , 0 ) , ( −6 , 0 ) , ( 0 , 3 ) , ( 0 , −3 ) ; foci: ( 3 3 , 0 ) , ( 3 3 , 0 )

Got questions? Get instant answers now!

( x 4 ) 2 25 + ( y + 3 ) 2 49 = 1

Got questions? Get instant answers now!

4 x 2 + y 2 + 16 x + 4 y 44 = 0

center: ( −2 , −2 ) ; vertices: ( 2 , −2 ) , ( −6 , −2 ) , ( −2 , 6 ) , ( −2 , −10 ) ; foci: ( −2 , −2 + 4 3 , ) , ( −2 , −2 −4 3 )

Got questions? Get instant answers now!

2 x 2 + 3 y 2 20 x + 12 y + 38 = 0

Got questions? Get instant answers now!

For the following exercises, use the given information to find the equation for the ellipse.

Center at ( 0 , 0 ) , focus at ( 3 , 0 ) , vertex at ( −5 , 0 )

x 2 25 + y 2 16 = 1

Got questions? Get instant answers now!

Center at ( 2 , −2 ) , vertex at ( 7 , −2 ) , focus at ( 4 , −2 )

Got questions? Get instant answers now!

A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?

Approximately 35.71 feet

Got questions? Get instant answers now!

The Hyperbola

For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.

( y + 1 ) 2 16 ( x 4 ) 2 36 = 1

( y + 1 ) 2 4 2 ( x 4 ) 2 6 2 = 1 ; center: ( 4 , −1 ) ; vertices: ( 4 , 3 ) , ( 4 , −5 ) ; foci: ( 4 , −1 + 2 13 ) , ( 4 , −1 2 13 )

Got questions? Get instant answers now!

9 y 2 4 x 2 + 54 y 16 x + 29 = 0

Got questions? Get instant answers now!

3 x 2 y 2 12 x 6 y 9 = 0

( x 2 ) 2 2 2 ( y + 3 ) 2 ( 2 3 ) 2 = 1 ; center: ( 2 , −3 ) ; vertices: ( 4 , −3 ) , ( 0 , −3 ) ; foci: ( 6 , −3 ) , ( −2 , −3 )

Got questions? Get instant answers now!

For the following exercises, graph the hyperbola, labeling vertices and foci.

Questions & Answers

what is mutation
Janga Reply
what is a cell
Sifune Reply
how is urine form
Sifune
what is antagonism?
mahase Reply
classification of plants, gymnosperm features.
Linsy Reply
what is the features of gymnosperm
Linsy
how many types of solid did we have
Samuel Reply
what is an ionic bond
Samuel
What is Atoms
Daprince Reply
what is fallopian tube
Merolyn
what is bladder
Merolyn
what's bulbourethral gland
Eduek Reply
urine is formed in the nephron of the renal medulla in the kidney. It starts from filtration, then selective reabsorption and finally secretion
onuoha Reply
State the evolution relation and relevance between endoplasmic reticulum and cytoskeleton as it relates to cell.
Jeremiah
what is heart
Konadu Reply
how is urine formed in human
Konadu
how is urine formed in human
Rahma
what is the diference between a cavity and a canal
Pelagie Reply
what is the causative agent of malaria
Diamond
malaria is caused by an insect called mosquito.
Naomi
Malaria is cause by female anopheles mosquito
Isaac
Malaria is caused by plasmodium Female anopheles mosquitoe is d carrier
Olalekan
a canal is more needed in a root but a cavity is a bad effect
Commander
what are pathogens
Don Reply
In biology, a pathogen (Greek: πάθος pathos "suffering", "passion" and -γενής -genēs "producer of") in the oldest and broadest sense, is anything that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a germ. The term pathogen came into use in the 1880s.[1][2
Zainab
A virus
Commander
Definition of respiration
Muhsin Reply
respiration is the process in which we breath in oxygen and breath out carbon dioxide
Achor
how are lungs work
Commander
where does digestion begins
Achiri Reply
in the mouth
EZEKIEL
what are the functions of follicle stimulating harmones?
Rashima Reply
stimulates the follicle to release the mature ovum into the oviduct
Davonte
what are the functions of Endocrine and pituitary gland
Chinaza
endocrine secrete hormone and regulate body process
Achor
while pituitary gland is an example of endocrine system and it's found in the Brain
Achor
what's biology?
Egbodo Reply
Biology is the study of living organisms, divided into many specialized field that cover their morphology, physiology,anatomy, behaviour,origin and distribution.
Lisah
biology is the study of life.
Alfreda
Biology is the study of how living organisms live and survive in a specific environment
Sifune
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask