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

( x 2 ) 2 7 2 + ( y 4 ) 2 5 2 = 1 ; Endpoints of major axis ( 9 , 4 ) , ( 5 , 4 ) . Endpoints of minor axis ( 2 , 9 ) , ( 2 , 1 ) . Foci at ( 2 + 2 6 , 4 ) , ( 2 2 6 , 4 ) .

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( x 2 ) 2 81 + ( y + 1 ) 2 16 = 1

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( x + 5 ) 2 4 + ( y 7 ) 2 9 = 1

( x + 5 ) 2 2 2 + ( y 7 ) 2 3 2 = 1 ; Endpoints of major axis ( 5 , 10 ) , ( 5 , 4 ) . Endpoints of minor axis ( 3 , 7 ) , ( 7 , 7 ) . Foci at ( 5 , 7 + 5 ) , ( 5 , 7 5 ) .

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( x 7 ) 2 49 + ( y 7 ) 2 49 = 1

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4 x 2 8 x + 9 y 2 72 y + 112 = 0

( x 1 ) 2 3 2 + ( y 4 ) 2 2 2 = 1 ; Endpoints of major axis ( 4 , 4 ) , ( 2 , 4 ) . Endpoints of minor axis ( 1 , 6 ) , ( 1 , 2 ) . Foci at ( 1 + 5 , 4 ) , ( 1 5 , 4 ) .

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9 x 2 54 x + 9 y 2 54 y + 81 = 0

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4 x 2 24 x + 36 y 2 360 y + 864 = 0

( x 3 ) 2 ( 3 2 ) 2 + ( y 5 ) 2 ( 2 ) 2 = 1 ; Endpoints of major axis ( 3 + 3 2 , 5 ) , ( 3 3 2 , 5 ) . Endpoints of minor axis ( 3 , 5 + 2 ) , ( 3 , 5 2 ) . Foci at ( 7 , 5 ) , ( 1 , 5 ) .

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4 x 2 + 24 x + 16 y 2 128 y + 228 = 0

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4 x 2 + 40 x + 25 y 2 100 y + 100 = 0

( x + 5 ) 2 ( 5 ) 2 + ( y 2 ) 2 ( 2 ) 2 = 1 ; Endpoints of major axis ( 0 , 2 ) , ( 10 , 2 ) . Endpoints of minor axis ( 5 , 4 ) , ( 5 , 0 ) . Foci at ( 5 + 21 , 2 ) , ( 5 21 , 2 ) .

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x 2 + 2 x + 100 y 2 1000 y + 2401 = 0

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4 x 2 + 24 x + 25 y 2 + 200 y + 336 = 0

( x + 3 ) 2 ( 5 ) 2 + ( y + 4 ) 2 ( 2 ) 2 = 1 ; Endpoints of major axis ( 2 , 4 ) , ( 8 , 4 ) . Endpoints of minor axis ( 3 , 2 ) , ( 3 , 6 ) . Foci at ( 3 + 21 , 4 ) , ( 3 21 , 4 ) .

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9 x 2 + 72 x + 16 y 2 + 16 y + 4 = 0

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For the following exercises, find the foci for the given ellipses.

( x + 3 ) 2 25 + ( y + 1 ) 2 36 = 1

Foci ( 3 , 1 + 11 ) , ( 3 , 1 11 )

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( x + 1 ) 2 100 + ( y 2 ) 2 4 = 1

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x 2 + y 2 = 1

Focus ( 0 , 0 )

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x 2 + 4 y 2 + 4 x + 8 y = 1

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10 x 2 + y 2 + 200 x = 0

Foci ( 10 , 30 ) , ( 10 , 30 )

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Graphical

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

x 2 16 + y 2 9 = 1

Center ( 0 , 0 ) , Vertices ( 4 , 0 ) , ( 4 , 0 ) , ( 0 , 3 ) , ( 0 , 3 ) , Foci ( 7 , 0 ) , ( 7 , 0 )

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81 x 2 + 49 y 2 = 1

Center ( 0 , 0 ) , Vertices ( 1 9 , 0 ) , ( 1 9 , 0 ) , ( 0 , 1 7 ) , ( 0 , 1 7 ) , Foci ( 0 , 4 2 63 ) , ( 0 , 4 2 63 )

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( x 2 ) 2 64 + ( y 4 ) 2 16 = 1

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( x + 3 ) 2 9 + ( y 3 ) 2 9 = 1

Center ( 3 , 3 ) , Vertices ( 0 , 3 ) , ( 6 , 3 ) , ( 3 , 0 ) , ( 3 , 6 ) , Focus ( 3 , 3 )

Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

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x 2 2 + ( y + 1 ) 2 5 = 1

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4 x 2 8 x + 16 y 2 32 y 44 = 0

Center ( 1 , 1 ) , Vertices ( 5 , 1 ) , ( 3 , 1 ) , ( 1 , 3 ) , ( 1 , 1 ) , Foci ( 1 , 1 + 4 3 ) , ( 1 , 1 4 3 )

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x 2 8 x + 25 y 2 100 y + 91 = 0

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x 2 + 8 x + 4 y 2 40 y + 112 = 0

Center ( 4 , 5 ) , Vertices ( 2 , 5 ) , ( 6 , 4 ) , ( 4 , 6 ) , ( 4 , 4 ) , Foci ( 4 + 3 , 5 ) , ( 4 3 , 5 )

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64 x 2 + 128 x + 9 y 2 72 y 368 = 0

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16 x 2 + 64 x + 4 y 2 8 y + 4 = 0

Center ( 2 , 1 ) , Vertices ( 0 , 1 ) , ( 4 , 1 ) , ( 2 , 5 ) , ( 2 , 3 ) , Foci ( 2 , 1 + 2 3 ) , ( 2 , 1 2 3 )

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100 x 2 + 1000 x + y 2 10 y + 2425 = 0

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

Center ( 2 , 2 ) , Vertices ( 0 , 2 ) , ( 4 , 2 ) , ( 2 , 0 ) , ( 2 , 4 ) , Focus ( 2 , 2 )

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For the following exercises, use the given information about the graph of each ellipse to determine its equation.

Center at the origin, symmetric with respect to the x - and y -axes, focus at ( 4 , 0 ) , and point on graph ( 0 , 3 ) .

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Center at the origin, symmetric with respect to the x - and y -axes, focus at ( 0 , −2 ) , and point on graph ( 5 , 0 ) .

x 2 25 + y 2 29 = 1

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Center at the origin, symmetric with respect to the x - and y -axes, focus at ( 3 , 0 ) , and major axis is twice as long as minor axis.

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Center ( 4 , 2 ) ; vertex ( 9 , 2 ) ; one focus: ( 4 + 2 6 , 2 ) .

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

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Center ( 3 , 5 ) ; vertex ( 3 , 11 ) ; one focus: ( 3 ,  5+4 2 )

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Center ( −3 , 4 ) ; vertex ( 1 , 4 ) ; one focus: ( −3 + 2 3 , 4 )

( x + 3 ) 2 16 + ( y 4 ) 2 4 = 1

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For the following exercises, given the graph of the ellipse, determine its equation.

( x + 2 ) 2 4 + ( y 2 ) 2 9 = 1

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Extensions

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula Area = a b π .

( x 3 ) 2 9 + ( y 3 ) 2 16 = 1

Area = 12π square units

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( x + 6 ) 2 16 + ( y 6 ) 2 36 = 1

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( x + 1 ) 2 4 + ( y 2 ) 2 5 = 1

Area = 2 5 π square units

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4 x 2 8 x + 9 y 2 72 y + 112 = 0

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9 x 2 54 x + 9 y 2 54 y + 81 = 0

Area = 9π square units

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Real-world applications

Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.

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Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of h , the height.

x 2 4 h 2 + y 2 1 4 h 2 = 1

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An arch has the shape of a semi-ellipse (the top half of an ellipse). The arch has a height of 8 feet and a span of 20 feet. Find an equation for the ellipse, and use that to find the height to the nearest 0.01 foot of the arch at a distance of 4 feet from the center.

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An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.

x 2 400 + y 2 144 = 1 . Distance = 17.32 feet

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A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.

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A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.

Approximately 51.96 feet

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A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus, and the other focus is 80 feet away, what is the length and height at the center of the gallery?

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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
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what is viscosity?
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emma Reply
what is chemistry
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what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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