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Suppose a whispering chamber is 480 feet long and 320 feet wide.

  1. What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point ( 0 , 0 ) .
  2. If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot.
  1. x 2 57 , 600 + y 2 25 , 600 = 1
  2. The people are standing 358 feet apart.
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Access these online resources for additional instruction and practice with ellipses.

Key equations

Horizontal ellipse, center at origin x 2 a 2 + y 2 b 2 = 1 ,   a > b
Vertical ellipse, center at origin x 2 b 2 + y 2 a 2 = 1 ,   a > b
Horizontal ellipse, center ( h , k ) ( x h ) 2 a 2 + ( y k ) 2 b 2 = 1 ,   a > b
Vertical ellipse, center ( h , k ) ( x h ) 2 b 2 + ( y k ) 2 a 2 = 1 ,   a > b

Key concepts

  • An ellipse is the set of all points ( x , y ) in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).
  • When given the coordinates of the foci and vertices of an ellipse, we can write the equation of the ellipse in standard form. See [link] and [link] .
  • When given an equation for an ellipse centered at the origin in standard form, we can identify its vertices, co-vertices, foci, and the lengths and positions of the major and minor axes in order to graph the ellipse. See [link] and [link] .
  • When given the equation for an ellipse centered at some point other than the origin, we can identify its key features and graph the ellipse. See [link] and [link] .
  • Real-world situations can be modeled using the standard equations of ellipses and then evaluated to find key features, such as lengths of axes and distance between foci. See [link] .

Section exercises

Verbal

Define an ellipse in terms of its foci.

An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

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Where must the foci of an ellipse lie?

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What special case of the ellipse do we have when the major and minor axis are of the same length?

This special case would be a circle.

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For the special case mentioned above, what would be true about the foci of that ellipse?

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What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the y -axis?

It is symmetric about the x -axis, y -axis, and the origin.

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Algebraic

For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.

4 x 2 + 9 y 2 = 36

yes; x 2 3 2 + y 2 2 2 = 1

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4 x 2 + 9 y 2 = 1

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

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

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For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.

x 2 4 + y 2 49 = 1

x 2 2 2 + y 2 7 2 = 1 ; Endpoints of major axis ( 0 , 7 ) and ( 0 , 7 ) . Endpoints of minor axis ( 2 , 0 ) and ( 2 , 0 ) . Foci at ( 0 , 3 5 ) , ( 0 , 3 5 ) .

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x 2 + 9 y 2 = 1

x 2 ( 1 ) 2 + y 2 ( 1 3 ) 2 = 1 ; Endpoints of major axis ( 1 , 0 ) and ( 1 , 0 ) . Endpoints of minor axis ( 0 , 1 3 ) , ( 0 , 1 3 ) . Foci at ( 2 2 3 , 0 ) , ( 2 2 3 , 0 ) .

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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
Natural numbers of definitions
Ali
(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
0.2502502
Ali
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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