12.1 The ellipse  (Page 8/16)

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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 $\text{\hspace{0.17em}}\left(0,0\right).$
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 $\text{\hspace{0.17em}}\left(\frac{2\sqrt{2}}{3},0\right),\left(-\frac{2\sqrt{2}}{3},0\right).$

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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