<< Chapter < Page Chapter >> Page >

( 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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

x 2 + y 2 = 1

Focus ( 0 , 0 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

10 x 2 + y 2 + 200 x = 0

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( 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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 )

Got questions? Get instant answers now!

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 ) .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Area = 2 5 π square units

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Area = 9π square units

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!
Practice Key Terms 7

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask