Volume of a Football A football is in the shape of a prolate spheroid, which is simply a

The distance d from to is _____.

To complete the square of add .

Find the intercepts of the equation y2 = 16 - 4x2.

The point that is symmetric with respect to the y-axis to the point is _____.

To graph shift the graph of to the (left/right) _____ unit(s) and then (up/down) _____ unit(s).

The standard equation of a circle with center at and radius 1 is _____.

A(n) is the collection of all points in the plane the sum of whose distances from two fixed points is a constant.

For an ellipse, the foci lie on a line called the axis.

For the ellipse the vertices are the points and .

For the ellipse the value of a is , the value of b is , and the major axis is the -axis.

If the center of an ellipse is the major axis is parallel to the x-axis, and the distance from the center of the ellipse to its vertices is units, then the coordinates of the vertices are and .

If the foci of an ellipse are and , then the coordinates of the center of the ellipse are .

In Problems 1316, the graph of an ellipse is given. Match each graph to its equation.

In Problems 1316, the graph of an ellipse is given. Match each graph to its equation.

In Problems 1316, the graph of an ellipse is given. Match each graph to its equation.

In Problems 1316, the graph of an ellipse is given. Match each graph to its equation.

x2 25 + y2 4 = 1

x2 9 + y2 4 = 1

x2 9 + y2 25 = 1

x2 + y2 16 = 1

4x2 + y2 = 16

x2 + 9y2 = 18

4y2 + x2 = 8

4y2 + 9x2 = 36

x2 + y2 = 16

x2 + y2 = 4

Center at focus at vertex at 10, 02; 13, 02; 15, 02

Center at focus at vertex at 10, 02; 1-1, 02; 13, 02

Center at focus at vertex at 10, 02; 10, -42; 10, 52

Center at focus at vertex at 10, 02; 10, 12; 10, -22

Foci at length of the major axis is 6

Foci at length of the major axis is 8

Focus at vertices at 1-4, 02; 1;5, 02

Focus at vertices at 10, -42; 10, ;82

Foci at 10, ;32; x-intercepts are ;2

Vertices at 1;4, 02; y-intercepts are ;1

Center at vertex at 10, 02; 10, 42; b = 1

Vertices at 1;5, 02; c = 2

In Problems 3942, write an equation for each ellipse. x y 3 3 3 3 (1, 1) (1, 1) x y 3 3

In Problems 3942, write an equation for each ellipse. x y 3 3 3 3 (1, 1) (1, 1) x y 3 4

In Problems 3942, write an equation for each ellipse. x y 3 3 3 3 (1, 1) (1, 1) x y 3 5

In Problems 3942, write an equation for each ellipse. x y 3 3 3 3 (1, 1) (1, 1) x y 3 6

1x - 322 4 + 1y + 122 9 = 1

1x + 422 9 + 1y + 222 4 = 1

1x + 522 + 41y - 422 = 16

91x - 322 + 1y + 222 = 18

x2 + 4x + 4y2 - 8y + 4 = 0

x2 + 3y2 - 12y + 9 = 0

2x2 + 3y2 - 8x + 6y + 5 = 0 5

4x2 + 3y2 + 8x - 6y = 5

9x2 + 4y2 - 18x + 16y - 11 = 0

x2 + 9y2 + 6x - 18y + 9 = 0

4x2 + y2 + 4y = 0

9x2 + y2 - 18x = 0

Center at vertex at focus at 12, -22; 17, -22; 14, -22

Center at vertex at focus at 1-3, 12; 1-3, 32; 1-3, 02

Vertices at and focus at 14, 32 14, 92; 14, 82

Foci at and vertex at 11, 22 1-3, 22; 1-4, 22

Foci at and length of the major axis is 8

Vertices at and 12, 52 12, -12; c = 2

Center at focus at contains the point 11, 22; 14, 22; 11, 32

Center at focus at contains the point 11, 22; 11, 42; 12, 22

Center at vertex at contains the point 11, 22; 14, 22; 11, 52

Center at vertex at contains the point 11 + 23, 32

f1x2 = 416 - 4x2

f1x2 = 49 - 9x2

f1x2 = -464 - 16x2 6

f1x2 = -464 - 16x2 6

Semielliptical Arch Bridge An arch in the shape of the upper half of an ellipse is used to support a bridge that is to span a river 20 meters wide. The center of the arch is 6 meters above the center of the river. See the figure.Write an equation for the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of the arch.

Semielliptical Arch Bridge The arch of a bridge is a semiellipse with a horizontal major axis. The span is 30 feet, and the top of the arch is 10 feet above the major axis. The roadway is horizontal and is 2 feet above the top of the arch. Find the vertical distance from the roadway to the arch at 5-foot intervals along the roadway.

Whispering Gallery A hall 100 feet in length is to be designed as a whispering gallery. If the foci are located 25 feet from the center, how high will the ceiling be at the center?

Whispering Gallery Jim, standing at one focus of a whispering gallery, is 6 feet from the nearest wall. His friend is standing at the other focus, 100 feet away. What is the length of this whispering gallery? How high is its elliptical ceiling at the center?

Semielliptical Arch Bridge A bridge is built in the shape of a semielliptical arch. The bridge has a span of 120 feet and a maximum height of 25 feet. Choose a suitable rectangular coordinate system and find the height of the arch at distances of 10, 30, and 50 feet from the center.

Semielliptical Arch Bridge A bridge is to be built in the shape of a semielliptical arch and is to have a span of 100 feet. The height of the arch, at a distance of 40 feet from the center, is to be 10 feet. Find the height of the arch at its center

Racetrack Design Consult the figure. A racetrack is in the shape of an ellipse, 100 feet long and 50 feet wide.What is the width 10 feet from a vertex?

Semielliptical Arch Bridge An arch for a bridge over a highway is in the form of half an ellipse.The top of the arch is 20 feet above the ground level (the major axis). The highway has four lanes, each 12 feet wide; a center safety strip 8 feet wide; and two side strips, each 4 feet wide. What should the span of the bridge be (the length of its major axis) if the height 28 feet from the center is to be 13 feet?

Installing a Vent Pipe A homeowner is putting in a fireplace that has a 4-inch-radius vent pipe. He needs to cut an elliptical hole in his roof to accommodate the pipe. If the pitch of his roof is , (a rise of 5, run of 4) what are the dimensions of the hole?

Volume of a Football A football is in the shape of a prolate spheroid, which is simply a solid obtained by rotating an ellipse about its major axis.An inflated NFL football averages 11.125 inches in length and 28.25 inches in center circumference. If the volume of a prolate spheroid is , how much air does the football contain? (Neglect material thickness).

Earth The mean distance of Earth from the Sun is 93 million miles. If the aphelion of Earth is 94.5 million miles, what is the perihelion? Write an equation for the orbit of Earth around the Sun.

Mars The mean distance of Mars from the Sun is 142 million miles. If the perihelion of Mars is 128.5 million miles, what is the aphelion? Write an equation for the orbit of Mars about the Sun.

Jupiter The aphelion of Jupiter is 507 million miles. If the distance from the center of its elliptical orbit to the Sun is 23.2 million miles, what is the perihelion? What is the mean distance? Write an equation for the orbit of Jupiter around the Sun.

Pluto The perihelion of Pluto is 4551 million miles, and the distance from the center of its elliptical orbit to the Sun is 897.5 million miles. Find the aphelion of Pluto. What is the mean distance of Pluto from the Sun? Write an equation for the orbit of Pluto about the Sun.

Show that an equation of the form Ax2 + Cy2 + F = 0, A Z 0, C Z 0, F Z 0 where A and C are of the same sign and F is of opposite sign, (a) Is the equation of an ellipse with center at if (b) Is the equation of a circle with center if 10, 02 A = C.

Show that the graph of an equation of the form where A and C are of the same sign, (a) Is an ellipse if is the same sign as A. D2 4A + E2 4C - F Ax2 + Cy2 + Dx + Ey + F = 0, A Z 0, C Z 0 (b) Is a point if (c) Contains no points if is of opposite sign to A. D2 4A + E2 4C - F D2 4A + E2 4C - F = 0.

The eccentricity e of an ellipse is defined as the number where a is the distance of a vertex from the center and c is the distance of a focus from the center. Because it follows that Write a brief paragraph about the general shape of each of the following ellipses. Be sure to justify your conclusions. (a) Eccentricity close to 0 (b) (c) Eccentricity close to 1