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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

use the properties of logarithms to rewrite as a single logarithmic expression 2+log base 5 t divided by log base 5 17
lucas Reply
(cos²x +tan²x)÷(sec²x+cos²x)
Sonu Reply
Solve the problem 3x^2_2x+4=0
Ayoman Reply
can anyone recommend an app or. website to help me refresh my knowledge and abilities? I stopped learning more about calc and trig in the 90's
scott Reply
if 6x=-2 find value of 6/x
issaya Reply
l don't know
Aman
I don't know
Aman
l don't know
Aman
I don't know
Aman
x=.333333333334, so 6/x= 18±
scott
x=.33333333334± 6/x=18±
scott
that is strange, I don't remember inserting the À. the result I got was x= 3333333334± 6/x=18±
scott
I need glasses
scott
X=(-1÷3) X equals minus one third
Melvin
how can I get the answer if don't know it
Adamma
x=-3
Dani
-18
Nwaneri
if sin15°=√p, express the following in terms of p
Norman Reply
I am learning and does someone have the time to tell me what or where this particular equation would be used?
scott
prove sin²x+cos²x=3+cos4x
Kiddy Reply
the difference between two signed numbers is -8.if the minued is 5,what is the subtrahend
jeramie Reply
the difference between two signed numbers is -8.if the minuend is 5.what is the subtrahend
jeramie
what are odd numbers
micheal Reply
numbers that leave a remainder when divided by 2
Thorben
1,3,5,7,... 99,...867
Thorben
7%2=1, 679%2=1, 866245%2=1
Thorben
the third and the seventh terms of a G.P are 81 and 16, find the first and fifth terms.
Suleiman Reply
if a=3, b =4 and c=5 find the six trigonometric value sin
Martin Reply
ask
Ans
pls how do I factorize x⁴+x³-7x²-x+6=0
Gift Reply
in a function the input value is called
Rimsha Reply
how do I test for values on the number line
Modesta Reply
if a=4 b=4 then a+b=
Rimsha Reply
a+b+2ab
Kin
commulative principle
DIOSDADO
a+b= 4+4=8
Mimi
If a=4 and b=4 then we add the value of a and b i.e a+b=4+4=8.
Tariq
Practice Key Terms 7

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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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