10.1 The ellipse  (Page 7/16)

 Page 7 / 16

Graphing an ellipse centered at ( h , k ) by first writing it in standard form

Graph the ellipse given by the equation $\text{\hspace{0.17em}}4{x}^{2}+9{y}^{2}-40x+36y+100=0.\text{\hspace{0.17em}}$ Identify and label the center, vertices, co-vertices, and foci.

We must begin by rewriting the equation in standard form.

$4{x}^{2}+9{y}^{2}-40x+36y+100=0$

Group terms that contain the same variable, and move the constant to the opposite side of the equation.

$\left(4{x}^{2}-40x\right)+\left(9{y}^{2}+36y\right)=-100$

Factor out the coefficients of the squared terms.

$4\left({x}^{2}-10x\right)+9\left({y}^{2}+4y\right)=-100$

Complete the square twice. Remember to balance the equation by adding the same constants to each side.

$4\left({x}^{2}-10x+25\right)+9\left({y}^{2}+4y+4\right)=-100+100+36$

Rewrite as perfect squares.

$4{\left(x-5\right)}^{2}+9{\left(y+2\right)}^{2}=36$

Divide both sides by the constant term to place the equation in standard form.

$\frac{{\left(x-5\right)}^{2}}{9}+\frac{{\left(y+2\right)}^{2}}{4}=1$

Now that the equation is in standard form, we can determine the position of the major axis. Because $\text{\hspace{0.17em}}9>4,\text{\hspace{0.17em}}$ the major axis is parallel to the x -axis. Therefore, the equation is in the form $\text{\hspace{0.17em}}\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}{a}^{2}=9\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{b}^{2}=4.\text{\hspace{0.17em}}$ It follows that:

• the center of the ellipse is $\text{\hspace{0.17em}}\left(h,k\right)=\left(5,-2\right)$
• the coordinates of the vertices are $\text{\hspace{0.17em}}\left(h±a,k\right)=\left(5±\sqrt{9},-2\right)=\left(5±3,-2\right),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(2,-2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8,-2\right)$
• the coordinates of the co-vertices are $\text{\hspace{0.17em}}\left(h,k±b\right)=\left(\text{5},-2±\sqrt{4}\right)=\left(\text{5},-2±2\right),\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\left(5,-4\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,\text{0}\right)$
• the coordinates of the foci are $\text{\hspace{0.17em}}\left(h±c,k\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.\text{\hspace{0.17em}}$ Solving for $\text{\hspace{0.17em}}c,\text{\hspace{0.17em}}$ we have:
$\begin{array}{l}c=±\sqrt{{a}^{2}-{b}^{2}}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=±\sqrt{9-4}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=±\sqrt{5}\hfill \end{array}$

Therefore, the coordinates of the foci are $\text{\hspace{0.17em}}\left(\text{5}-\sqrt{5},-2\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(\text{5+}\sqrt{5},-2\right).$

Next we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse as shown in [link] .

Express the equation of the ellipse given in standard form. Identify the center, vertices, co-vertices, and foci of the ellipse.

$4{x}^{2}+{y}^{2}-24x+2y+21=0$

$\text{\hspace{0.17em}}\frac{{\left(x-3\right)}^{2}}{4}+\frac{{\left(y+1\right)}^{2}}{16}=1;\text{\hspace{0.17em}}$ center: $\text{\hspace{0.17em}}\left(3,-1\right);\text{\hspace{0.17em}}$ vertices: $\text{\hspace{0.17em}}\left(3,-\text{5}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,\text{3}\right);\text{\hspace{0.17em}}$ co-vertices: $\text{\hspace{0.17em}}\left(1,-1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(5,-1\right);\text{\hspace{0.17em}}$ foci: $\text{\hspace{0.17em}}\left(3,-\text{1}-2\sqrt{3}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,-\text{1+}2\sqrt{3}\right)$

Solving applied problems involving ellipses

Many real-world situations can be represented by ellipses, including orbits of planets, satellites, moons and comets, and shapes of boat keels, rudders, and some airplane wings. A medical device called a lithotripter uses elliptical reflectors to break up kidney stones by generating sound waves. Some buildings, called whispering chambers, are designed with elliptical domes so that a person whispering at one focus can easily be heard by someone standing at the other focus. This occurs because of the acoustic properties of an ellipse. When a sound wave originates at one focus of a whispering chamber, the sound wave will be reflected off the elliptical dome and back to the other focus. See [link] . In the whisper chamber at the Museum of Science and Industry in Chicago, two people standing at the foci—about 43 feet apart—can hear each other whisper.

Locating the foci of a whispering chamber

The Statuary Hall in the Capitol Building in Washington, D.C. is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in [link] .

1. What is the standard form of the equation of the ellipse representing the outline of 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 senators standing at the foci of this room can hear each other whisper, how far apart are the senators? Round to the nearest foot.
1. We are assuming a horizontal ellipse with center $\text{\hspace{0.17em}}\left(0,0\right),$ so we need to find an equation of the form $\text{\hspace{0.17em}}\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a>b.\text{\hspace{0.17em}}$ We know that the length of the major axis, $\text{\hspace{0.17em}}2a,\text{\hspace{0.17em}}$ is longer than the length of the minor axis, $\text{\hspace{0.17em}}2b.\text{\hspace{0.17em}}$ So the length of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor axis.
• Solving for $\text{\hspace{0.17em}}a,$ we have $\text{\hspace{0.17em}}2a=96,$ so $\text{\hspace{0.17em}}a=48,$ and $\text{\hspace{0.17em}}{a}^{2}=2304.$
• Solving for $\text{\hspace{0.17em}}b,$ we have $\text{\hspace{0.17em}}2b=46,$ so $\text{\hspace{0.17em}}b=23,$ and $\text{\hspace{0.17em}}{b}^{2}=529.$

Therefore, the equation of the ellipse is $\text{\hspace{0.17em}}\frac{{x}^{2}}{2304}+\frac{{y}^{2}}{529}=1.$

2. To find the distance between the senators, we must find the distance between the foci, $\text{\hspace{0.17em}}\left(±c,0\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}{c}^{2}={a}^{2}-{b}^{2}.\text{\hspace{0.17em}}$ Solving for $\text{\hspace{0.17em}}c,$ we have:

The points $\text{\hspace{0.17em}}\left(±42,0\right)\text{\hspace{0.17em}}$ represent the foci. Thus, the distance between the senators is $\text{\hspace{0.17em}}2\left(42\right)=84\text{\hspace{0.17em}}$ feet.

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich