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

r = 4 1 + 3   sin   θ

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r = 2 5 3   sin   θ

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

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r = 8 3 2   cos   θ

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r = 3 2 + 5   cos   θ

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

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r = 4 2 + 2   sin   θ

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r = 3 8 8   cos   θ

64 y 2 = 48 x + 9

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r = 2 6 + 7   cos   θ

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r = 5 5 11   sin   θ

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

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r ( 5 + 2   cos   θ ) = 6

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r ( 2 cos   θ ) = 1

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

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r ( 2.5 2.5   sin   θ ) = 5

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r = 6 sec   θ 2 + 3   sec   θ

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

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r = 6 csc   θ 3 + 2   csc   θ

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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

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r = 10 5 4   sin   θ

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r = 3 1 + 2   cos   θ

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r = 8 4 5   cos   θ

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r = 3 4 4   cos   θ

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r = 6 3 + 2   sin   θ

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r ( 3 4 sin   θ ) = 9

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r ( 3 2 sin   θ ) = 6

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r ( 6 4 cos   θ ) = 5

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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

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Directrix: x = 4 ; e = 5

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Directrix: y = 2 ; e = 2

r = 4 1 + 2 sin θ

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Directrix: y = 2 ; e = 1 2

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Directrix: x = 1 ; e = 1

r = 1 1 + cos θ

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Directrix: x = 1 ; e = 1

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Directrix: x = 1 4 ; e = 7 2

r = 7 8 28 cos θ

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Directrix: y = 2 5 ; e = 7 2

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Directrix: y = 4 ; e = 3 2

r = 12 2 + 3 sin θ

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Directrix: x = −2 ; e = 8 3

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Directrix: x = −5 ; e = 3 4

r = 15 4 3 cos θ

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Directrix: y = 2 ; e = 2.5

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Directrix: x = −3 ; e = 1 3

r = 3 3 3 cos θ

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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

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

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

r = ± 2 4 cos θ + 3 sin θ

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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 )

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

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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 )

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

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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 )

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

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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 )

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

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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

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

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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

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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 )

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

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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 )

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

Questions & Answers

If c is the cost function for a particular product, find the marginal cost functions and their values at x=10 a. c(x) = 800+ 0.04x + 0.0002x² b. c(x) = 250 + 100x + 0.001x²
Mamush Reply
how can I find set theory
Ephraim Reply
how can I find set theory
Jarvis
is there an error on the one about the dime's thickness? says 2.2x10⁶=0.00135 m
Patrick Reply
hi, interested in algebra
Makan Reply
how to reduce an equation?
Makan
by manipulation of both side
Al
9(y+8)-27 is 9y+45. Why can't you reduce that to y+5? I know that's wrong but can't explain why
Patrick Reply
when you reduce an equation to its simplest terms, you can't change the value of the equation. reducing it to y + 5 is equivalent to dividing it by 9 which changes the value. you can multiply it by 1 or 9/9 which would give 9(y + 5). multiplying it by one does not change the value.
Philip
Given a polynomial expression, factor out the greatest common factor.
Hanu Reply
WHAT IS QUADRATIC EQUATION?
Charles Reply
WHAT IS SYSTEM OF LINEAR INEWUALITIES?
Charles
WHAT IS SYSTEM OF LINEAR INEWUALITIES?
Charles
complex perform
Angel
what is equation?
Charles Reply
what are equations?
Charles
Definition of economics according to karl Marx Thomas malthus Jeremy bentham David Ricardo J.K
Rakiya
Please help me is assignment
Rakiya
The 47th problem of Euclid
Kenneth
show that the set of all natural number form semi group under the composition of addition
Nikhil Reply
what is the meaning
Dominic
explain and give four Example hyperbolic function
Lukman Reply
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
Practice Key Terms 2

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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