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


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x 2 4 y 2 + 6 x + 32 y 91 = 0

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2 y 2 x 2 12 y 6 = 0


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For the following exercises, find the equation of the hyperbola.

Center at ( 0 , 0 ) , vertex at ( 0 , 4 ) , focus at ( 0 , −6 )

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Foci at ( 3 , 7 ) and ( 7 , 7 ) , vertex at ( 6 , 7 )

( x 5 ) 2 1 ( y 7 ) 2 3 = 1

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The Parabola

For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.

( x + 2 ) 2 = 1 2 ( y 1 )

( x + 2 ) 2 = 1 2 ( y 1 ) ; vertex: ( −2 , 1 ) ; focus: ( −2 , 9 8 ) ; directrix: y = 7 8

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y 2 6 y 6 x 3 = 0

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x 2 + 10 x y + 23 = 0

( x + 5 ) 2 = ( y + 2 ) ; vertex: ( 5 , 2 ) ; focus: ( 5 , 7 4 ) ; directrix: y = 9 4

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For the following exercises, graph the parabola, labeling vertex, focus, and directrix.

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


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x 2 8 x 10 y + 46 = 0

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2 y 2 + 12 y + 6 x + 15 = 0


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For the following exercises, write the equation of the parabola using the given information.

Focus at ( −4 , 0 ) ; directrix is x = 4

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Focus at ( 2 , 9 8 ) ; directrix is y = 7 8

( x 2 ) 2 = ( 1 2 ) ( y 1 )

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A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 5 feet across at its opening and 1.5 feet deep.

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Rotation of Axes

For the following exercises, determine which of the conic sections is represented.

16 x 2 + 24 x y + 9 y 2 + 24 x 60 y 60 = 0

B 2 4 A C = 0 , parabola

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4 x 2 + 14 x y + 5 y 2 + 18 x 6 y + 30 = 0

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4 x 2 + x y + 2 y 2 + 8 x 26 y + 9 = 0

B 2 4 A C = 31 < 0 , ellipse

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For the following exercises, determine the angle θ that will eliminate the x y term, and write the corresponding equation without the x y term.

x 2 + 4 x y 2 y 2 6 = 0

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x 2 x y + y 2 6 = 0

θ = 45 , x 2 + 3 y 2 12 = 0

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For the following exercises, graph the equation relative to the x y system in which the equation has no x y term.

9 x 2 24 x y + 16 y 2 80 x 60 y + 100 = 0

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x 2 x y + y 2 2 = 0

θ = 45

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6 x 2 + 24 x y y 2 12 x + 26 y + 11 = 0

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Conic Sections in Polar Coordinates

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

r = 10 1 5   cos   θ

Hyperbola with e = 5 and directrix 2 units to the left of the pole.

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r = 6 3 + 2   cos   θ

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r = 1 4 + 3   sin   θ

Ellipse with e = 3 4 and directrix 1 3 unit above the pole.

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r = 3 5 5   sin   θ

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

r = 8 4 + 3   sin   θ

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r = 10 4 + 5   cos   θ


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r = 9 3 6   cos   θ

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For the following exercises, given information about the graph of a conic with focus at the origin, find the equation in polar form.

Directrix is x = 3 and eccentricity e = 1

r = 3 1 + cos     θ

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Directrix is y = −2 and eccentricity e = 4

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Practice test

For the following exercises, write the equation in standard form and state the center, vertices, and foci.

x 2 9 + y 2 4 = 1

x 2 3 2 + y 2 2 2 = 1 ; center: ( 0 , 0 ) ; vertices: ( 3 , 0 ) , ( –3 , 0 ) , ( 0 , 2 ) , ( 0 , −2 ) ; foci: ( 5 , 0 ) , ( 5 , 0 )

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9 y 2 + 16 x 2 36 y + 32 x 92 = 0

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For the following exercises, sketch the graph, identifying the center, vertices, and foci.

( x 3 ) 2 64 + ( y 2 ) 2 36 = 1

center: ( 3 , 2 ) ; vertices: ( 11 , 2 ) , ( −5 , 2 ) , ( 3 , 8 ) , ( 3 , −4 ) ; foci: ( 3 + 2 7 , 2 ) , ( 3 2 7 , 2 )

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2 x 2 + y 2 + 8 x 6 y 7 = 0

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Write the standard form equation of an ellipse with a center at ( 1 , 2 ) , vertex at ( 7 , 2 ) , and focus at ( 4 , 2 ).

( x 1 ) 2 36 + ( y 2 ) 2 27 = 1

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A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?

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

x 2 49 y 2 81 = 1

x 2 7 2 y 2 9 2 = 1 ; center: ( 0 , 0 ) ; vertices ( 7 , 0 ) , ( −7 , 0 ) ; foci: ( 130 , 0 ) , ( 130 , 0 ) ; asymptotes: y = ± 9 7 x

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16 y 2 9 x 2 + 128 y + 112 = 0

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For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.

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

center: ( 3 , −3 ) ; vertices: ( 8 , −3 ) , ( −2 , −3 ) ; foci: ( 3 + 26 , −3 ) , ( 3 26 , −3 ) ; asymptotes: y = ± 1 5 ( x 3 ) 3

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y 2 x 2 + 4 y 4 x 18 = 0

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Write the standard form equation of a hyperbola with foci at ( 1 , 0 ) and ( 1 , 6 ) , and a vertex at ( 1 , 2 ) .

( y 3 ) 2 1 ( x 1 ) 2 8 = 1

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For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.

3 x 2 12 x y + 11 = 0

( x 2 ) 2 = 1 3 ( y + 1 ) ; vertex: ( 2 , −1 ) ; focus: ( 2 , 11 12 ) ; directrix: y = 13 12

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For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.

( x 1 ) 2 = −4 ( y + 3 )

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y 2 + 8 x 8 y + 40 = 0


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Write the equation of a parabola with a focus at ( 2 , 3 ) and directrix y = −1.

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A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?

Approximately 8.49 feet

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For the following exercises, determine which conic section is represented by the given equation, and then determine the angle θ that will eliminate the x y term.

3 x 2 2 x y + 3 y 2 = 4

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x 2 + 4 x y + 4 y 2 + 6 x 8 y = 0

parabola; θ 63.4

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For the following exercises, rewrite in the x y system without the x y term, and graph the rotated graph.

11 x 2 + 10 3 x y + y 2 = 4

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16 x 2 + 24 x y + 9 y 2 125 x = 0

x 2 4 x + 3 y = 0

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For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity.

r = 5 4 + 6   cos   θ

Hyperbola with e = 3 2 , and directrix 5 6 units to the right of the pole.

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

r = 12 4 8   sin   θ

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r = 2 4 + 4   sin   θ

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Find a polar equation of the conic with focus at the origin, eccentricity of e = 2 , and directrix: x = 3.

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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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