<< 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

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
Practice Key Terms 2

Get the best Algebra and trigonometry course in your pocket!





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