# 12.1 The ellipse  (Page 9/16)

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$4{x}^{2}+16{y}^{2}=1$

$\frac{{\left(x-2\right)}^{2}}{49}+\frac{{\left(y-4\right)}^{2}}{25}=1$

$\frac{{\left(x-2\right)}^{2}}{{7}^{2}}+\frac{{\left(y-4\right)}^{2}}{{5}^{2}}=1;\text{\hspace{0.17em}}$ Endpoints of major axis $\text{\hspace{0.17em}}\left(9,4\right),\left(-5,4\right).\text{\hspace{0.17em}}$ Endpoints of minor axis $\text{\hspace{0.17em}}\left(2,9\right),\left(2,-1\right).\text{\hspace{0.17em}}$ Foci at $\text{\hspace{0.17em}}\left(2+2\sqrt{6},4\right),\left(2-2\sqrt{6},4\right).$

$\frac{{\left(x-2\right)}^{2}}{81}+\frac{{\left(y+1\right)}^{2}}{16}=1$

$\frac{{\left(x+5\right)}^{2}}{4}+\frac{{\left(y-7\right)}^{2}}{9}=1$

$\frac{{\left(x+5\right)}^{2}}{{2}^{2}}+\frac{{\left(y-7\right)}^{2}}{{3}^{2}}=1;\text{\hspace{0.17em}}$ Endpoints of major axis $\text{\hspace{0.17em}}\left(-5,10\right),\left(-5,4\right).\text{\hspace{0.17em}}$ Endpoints of minor axis $\text{\hspace{0.17em}}\left(-3,7\right),\left(-7,7\right).\text{\hspace{0.17em}}$ Foci at $\text{\hspace{0.17em}}\left(-5,7+\sqrt{5}\right),\left(-5,7-\sqrt{5}\right).$

$\frac{{\left(x-7\right)}^{2}}{49}+\frac{{\left(y-7\right)}^{2}}{49}=1$

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

$\frac{{\left(x-1\right)}^{2}}{{3}^{2}}+\frac{{\left(y-4\right)}^{2}}{{2}^{2}}=1;\text{\hspace{0.17em}}$ Endpoints of major axis $\text{\hspace{0.17em}}\left(4,4\right),\left(-2,4\right).\text{\hspace{0.17em}}$ Endpoints of minor axis $\text{\hspace{0.17em}}\left(1,6\right),\left(1,2\right).\text{\hspace{0.17em}}$ Foci at $\text{\hspace{0.17em}}\left(1+\sqrt{5},4\right),\left(1-\sqrt{5},4\right).$

$9{x}^{2}-54x+9{y}^{2}-54y+81=0$

$4{x}^{2}-24x+36{y}^{2}-360y+864=0$

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

$4{x}^{2}+24x+16{y}^{2}-128y+228=0$

$4{x}^{2}+40x+25{y}^{2}-100y+100=0$

$\frac{{\left(x+5\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y-2\right)}^{2}}{{\left(2\right)}^{2}}=1;\text{\hspace{0.17em}}$ Endpoints of major axis $\text{\hspace{0.17em}}\left(0,2\right),\left(-10,2\right).\text{\hspace{0.17em}}$ Endpoints of minor axis $\text{\hspace{0.17em}}\left(-5,4\right),\left(-5,0\right).\text{\hspace{0.17em}}$ Foci at $\text{\hspace{0.17em}}\left(-5+\sqrt{21},2\right),\left(-5-\sqrt{21},2\right).$

${x}^{2}+2x+100{y}^{2}-1000y+2401=0$

$4{x}^{2}+24x+25{y}^{2}+200y+336=0$

$\frac{{\left(x+3\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y+4\right)}^{2}}{{\left(2\right)}^{2}}=1;\text{\hspace{0.17em}}$ Endpoints of major axis $\text{\hspace{0.17em}}\left(2,-4\right),\left(-8,-4\right).\text{\hspace{0.17em}}$ Endpoints of minor axis $\text{\hspace{0.17em}}\left(-3,-2\right),\left(-3,-6\right).\text{\hspace{0.17em}}$ Foci at $\text{\hspace{0.17em}}\left(-3+\sqrt{21},-4\right),\left(-3-\sqrt{21},-4\right).$

$9{x}^{2}+72x+16{y}^{2}+16y+4=0$

For the following exercises, find the foci for the given ellipses.

$\frac{{\left(x+3\right)}^{2}}{25}+\frac{{\left(y+1\right)}^{2}}{36}=1$

Foci $\text{\hspace{0.17em}}\left(-3,-1+\sqrt{11}\right),\left(-3,-1-\sqrt{11}\right)$

$\frac{{\left(x+1\right)}^{2}}{100}+\frac{{\left(y-2\right)}^{2}}{4}=1$

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

Focus $\text{\hspace{0.17em}}\left(0,0\right)$

${x}^{2}+4{y}^{2}+4x+8y=1$

$10{x}^{2}+{y}^{2}+200x=0$

Foci $\text{\hspace{0.17em}}\left(-10,30\right),\left(-10,-30\right)$

## Graphical

For the following exercises, graph the given ellipses, noting center, vertices, and foci.

$\frac{{x}^{2}}{25}+\frac{{y}^{2}}{36}=1$

$\frac{{x}^{2}}{16}+\frac{{y}^{2}}{9}=1$

Center $\text{\hspace{0.17em}}\left(0,0\right),\text{\hspace{0.17em}}$ Vertices $\text{\hspace{0.17em}}\left(4,0\right),\left(-4,0\right),\left(0,3\right),\left(0,-3\right),\text{\hspace{0.17em}}$ Foci $\text{\hspace{0.17em}}\left(\sqrt{7},0\right),\left(-\sqrt{7},0\right)$

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

$81{x}^{2}+49{y}^{2}=1$

Center $\text{\hspace{0.17em}}\left(0,0\right),\text{\hspace{0.17em}}$ Vertices $\text{\hspace{0.17em}}\left(\frac{1}{9},0\right),\left(-\frac{1}{9},0\right),\left(0,\frac{1}{7}\right),\left(0,-\frac{1}{7}\right),\text{\hspace{0.17em}}$ Foci $\text{\hspace{0.17em}}\left(0,\frac{4\sqrt{2}}{63}\right),\left(0,-\frac{4\sqrt{2}}{63}\right)$

$\frac{{\left(x-2\right)}^{2}}{64}+\frac{{\left(y-4\right)}^{2}}{16}=1$

$\frac{{\left(x+3\right)}^{2}}{9}+\frac{{\left(y-3\right)}^{2}}{9}=1$

Center $\text{\hspace{0.17em}}\left(-3,3\right),\text{\hspace{0.17em}}$ Vertices $\text{\hspace{0.17em}}\left(0,3\right),\left(-6,3\right),\left(-3,0\right),\left(-3,6\right),\text{\hspace{0.17em}}$ Focus $\text{\hspace{0.17em}}\left(-3,3\right)\text{\hspace{0.17em}}$

Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

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

$4{x}^{2}-8x+16{y}^{2}-32y-44=0$

Center $\text{\hspace{0.17em}}\left(1,1\right),\text{\hspace{0.17em}}$ Vertices $\text{\hspace{0.17em}}\left(5,1\right),\left(-3,1\right),\left(1,3\right),\left(1,-1\right),\text{\hspace{0.17em}}$ Foci $\text{\hspace{0.17em}}\left(1,1+4\sqrt{3}\right),\left(1,1-4\sqrt{3}\right)$

${x}^{2}-8x+25{y}^{2}-100y+91=0$

${x}^{2}+8x+4{y}^{2}-40y+112=0$

Center $\text{\hspace{0.17em}}\left(-4,5\right),\text{\hspace{0.17em}}$ Vertices $\text{\hspace{0.17em}}\left(-2,5\right),\left(-6,4\right),\left(-4,6\right),\left(-4,4\right),\text{\hspace{0.17em}}$ Foci $\text{\hspace{0.17em}}\left(-4+\sqrt{3},5\right),\left(-4-\sqrt{3},5\right)$

$64{x}^{2}+128x+9{y}^{2}-72y-368=0$

$16{x}^{2}+64x+4{y}^{2}-8y+4=0$

Center $\text{\hspace{0.17em}}\left(-2,1\right),\text{\hspace{0.17em}}$ Vertices $\text{\hspace{0.17em}}\left(0,1\right),\left(-4,1\right),\left(-2,5\right),\left(-2,-3\right),\text{\hspace{0.17em}}$ Foci $\text{\hspace{0.17em}}\left(-2,1+2\sqrt{3}\right),\left(-2,1-2\sqrt{3}\right)$

$100{x}^{2}+1000x+{y}^{2}-10y+2425=0$

$4{x}^{2}+16x+4{y}^{2}+16y+16=0$

Center $\text{\hspace{0.17em}}\left(-2,-2\right),\text{\hspace{0.17em}}$ Vertices $\text{\hspace{0.17em}}\left(0,-2\right),\left(-4,-2\right),\left(-2,0\right),\left(-2,-4\right),\text{\hspace{0.17em}}$ Focus $\text{\hspace{0.17em}}\left(-2,-2\right)$

For the following exercises, use the given information about the graph of each ellipse to determine its equation.

Center at the origin, symmetric with respect to the x - and y -axes, focus at $\text{\hspace{0.17em}}\left(4,0\right),$ and point on graph $\text{\hspace{0.17em}}\left(0,3\right).$

Center at the origin, symmetric with respect to the x - and y -axes, focus at $\text{\hspace{0.17em}}\left(0,-2\right),$ and point on graph $\text{\hspace{0.17em}}\left(5,0\right).$

$\frac{{x}^{2}}{25}+\frac{{y}^{2}}{29}=1$

Center at the origin, symmetric with respect to the x - and y -axes, focus at $\text{\hspace{0.17em}}\left(3,0\right),$ and major axis is twice as long as minor axis.

Center $\text{\hspace{0.17em}}\left(4,2\right)$ ; vertex $\text{\hspace{0.17em}}\left(9,2\right)$ ; one focus: $\text{\hspace{0.17em}}\left(4+2\sqrt{6},2\right)$ .

$\frac{{\left(x-4\right)}^{2}}{25}+\frac{{\left(y-2\right)}^{2}}{1}=1$

Center $\text{\hspace{0.17em}}\left(3,5\right)$ ; vertex $\text{\hspace{0.17em}}\left(3,11\right)$ ; one focus:

Center $\text{\hspace{0.17em}}\left(-3,4\right)$ ; vertex $\text{\hspace{0.17em}}\left(1,4\right)$ ; one focus: $\text{\hspace{0.17em}}\left(-3+2\sqrt{3},4\right)$

$\frac{{\left(x+3\right)}^{2}}{16}+\frac{{\left(y-4\right)}^{2}}{4}=1$

For the following exercises, given the graph of the ellipse, determine its equation.

$\frac{{x}^{2}}{81}+\frac{{y}^{2}}{9}=1$

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

## Extensions

For the following exercises, find the area of the ellipse. The area of an ellipse is given by the formula $\text{\hspace{0.17em}}\text{Area}=a\cdot b\cdot \pi .$

$\frac{{\left(x-3\right)}^{2}}{9}+\frac{{\left(y-3\right)}^{2}}{16}=1$

$\frac{{\left(x+6\right)}^{2}}{16}+\frac{{\left(y-6\right)}^{2}}{36}=1$

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

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

$9{x}^{2}-54x+9{y}^{2}-54y+81=0$

## Real-world applications

Find the equation of the ellipse that will just fit inside a box that is 8 units wide and 4 units high.

Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of $\text{\hspace{0.17em}}h,$ the height.

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

An arch has the shape of a semi-ellipse (the top half of an ellipse). The arch has a height of 8 feet and a span of 20 feet. Find an equation for the ellipse, and use that to find the height to the nearest 0.01 foot of the arch at a distance of 4 feet from the center.

An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.

$\frac{{x}^{2}}{400}+\frac{{y}^{2}}{144}=1$ . Distance = 17.32 feet

A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center.

A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.

Approximately 51.96 feet

A person is standing 8 feet from the nearest wall in a whispering gallery. If that person is at one focus, and the other focus is 80 feet away, what is the length and height at the center of the gallery?

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