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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

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
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
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Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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