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Suppose a whispering chamber is 480 feet long and 320 feet wide.
Access these online resources for additional instruction and practice with ellipses.
Horizontal ellipse, center at origin | $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1,\text{}ab$ |
Vertical ellipse, center at origin | $\frac{{x}^{2}}{{b}^{2}}+\frac{{y}^{2}}{{a}^{2}}=1,\text{}ab$ |
Horizontal ellipse, center $\text{\hspace{0.17em}}(h,k)$ | $\frac{{\left(x-h\right)}^{2}}{{a}^{2}}+\frac{{\left(y-k\right)}^{2}}{{b}^{2}}=1,\text{}ab$ |
Vertical ellipse, center $\text{\hspace{0.17em}}(h,k)$ | $\frac{{\left(x-h\right)}^{2}}{{b}^{2}}+\frac{{\left(y-k\right)}^{2}}{{a}^{2}}=1,\text{}ab$ |
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.
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 $\text{\hspace{0.17em}}\left(\frac{2\sqrt{2}}{3},0\right),\left(-\frac{2\sqrt{2}}{3},0\right).$
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