# 12.1 The ellipse  (Page 8/16)

 Page 8 / 16

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 $\text{\hspace{0.17em}}\left(0,0\right).$
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. $\frac{{x}^{2}}{57,600}+\frac{{y}^{2}}{25,600}=1$
2. The people are standing 358 feet apart.

Access these online resources for additional instruction and practice with ellipses.

## Key equations

 Horizontal ellipse, center at origin Vertical ellipse, center at origin Horizontal ellipse, center $\text{\hspace{0.17em}}\left(h,k\right)$ Vertical ellipse, center $\text{\hspace{0.17em}}\left(h,k\right)$

## Key concepts

• An ellipse is the set of all points $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ 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] .

## 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.

Where must the foci of an ellipse lie?

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.

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

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.

## Algebraic

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

$2{x}^{2}+y=4$

$4{x}^{2}+9{y}^{2}=36$

yes; $\text{\hspace{0.17em}}\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1$

$4{x}^{2}-{y}^{2}=4$

$4{x}^{2}+9{y}^{2}=1$

yes; $\frac{{x}^{2}}{{\left(\frac{1}{2}\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1$

$4{x}^{2}-8x+9{y}^{2}-72y+112=0$

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.

$\frac{{x}^{2}}{4}+\frac{{y}^{2}}{49}=1$

$\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{7}^{2}}=1;\text{\hspace{0.17em}}$ Endpoints of major axis $\text{\hspace{0.17em}}\left(0,7\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(0,-7\right).\text{\hspace{0.17em}}$ Endpoints of minor axis $\text{\hspace{0.17em}}\left(2,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-2,0\right).\text{\hspace{0.17em}}$ Foci at $\text{\hspace{0.17em}}\left(0,3\sqrt{5}\right),\left(0,-3\sqrt{5}\right).$

$\frac{{x}^{2}}{100}+\frac{{y}^{2}}{64}=1$

${x}^{2}+9{y}^{2}=1$

$\frac{{x}^{2}}{{\left(1\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1;\text{\hspace{0.17em}}$ Endpoints of major axis $\text{\hspace{0.17em}}\left(1,0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(-1,0\right).\text{\hspace{0.17em}}$ Endpoints of minor axis $\text{\hspace{0.17em}}\left(0,\frac{1}{3}\right),\left(0,-\frac{1}{3}\right).\text{\hspace{0.17em}}$ Foci at $\text{\hspace{0.17em}}\left(\frac{2\sqrt{2}}{3},0\right),\left(-\frac{2\sqrt{2}}{3},0\right).$

what are you up to?
nothing up todat yet
Miranda
hi
jai
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jai
Miranda Drice
jai
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jai
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Miranda
I am living in india
jai
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
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jai
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jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
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Miranda
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Jeffrey
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Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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Miranda
gotta practice, holmie
Steve
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Steve
I don't know why. But Im trying to like it.
Jeffrey
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Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
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Ye
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Nokwanda
C'est comment
Abdel
Hi
Amanda
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SORIE
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Ranjay
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ANSHU
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Chinni
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Hassan
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SORIE
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Abdel
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SORIE
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Yaona
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SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function