Solution: In the LORAN (LOng RAnge Navigation) radio navigation system, two radio

18 Find the vertex, focus, and directrix of the parabola and sketch its graph.x 2 6y

18 Find the vertex, focus, and directrix of the parabola and sketch its graph.2y 2 5x

18 Find the vertex, focus, and directrix of the parabola and sketch its graph.2x y 2 3x

18 Find the vertex, focus, and directrix of the parabola and sketch its graph.3x 2 8y 0 x

18 Find the vertex, focus, and directrix of the parabola and sketch its graph.x 2 2 8 y 3 x 1

18 Find the vertex, focus, and directrix of the parabola and sketch its graph.x 1 y 5 2 y 2

18 Find the vertex, focus, and directrix of the parabola and sketch its graph.y 2 2y 12x 25 0 y 1

18 Find the vertex, focus, and directrix of the parabola and sketch its graph.y 12x 2x 2 16 y

910 Find an equation of the parabola. Then find the focus and directrix.

910 Find an equation of the parabola. Then find the focus and directrix.

1116 Find the vertices and foci of the ellipse and sketch its graph.x 2 2 y 2 4 1 x

1116 Find the vertices and foci of the ellipse and sketch its graph.x 2 36 y 2 8 1 x

1116 Find the vertices and foci of the ellipse and sketch its graph.x 2 9y 2 9 10

1116 Find the vertices and foci of the ellipse and sketch its graph.100x 2 36y 2 225 9

1116 Find the vertices and foci of the ellipse and sketch its graph.9x 2 18x 4y 2 27 x

1116 Find the vertices and foci of the ellipse and sketch its graph.x 2 3y2 2x 12y 10 0 y x

1718 Find an equation of the ellipse. Then find its foci.

1718 Find an equation of the ellipse. Then find its foci.

1924 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.y 2 25 x 2 9 1 x

1924 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.x 2 36 y 2 64 1 x

1924 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.x 2 y 2 100 y

1924 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.y 2 16x 2 16 2

1924 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.4x 2 y2 24x 4y 28 0 10.5

1924 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph.y2 4x 2 2y 16x 31 4x 2

2530 Identify the type of conic section whose equation is given and find the vertices and foci.x 2 y 1 x

2530 Identify the type of conic section whose equation is given and find the vertices and foci.x 2 y 2 1 x

2530 Identify the type of conic section whose equation is given and find the vertices and foci.x 2 4y 2y 2 y

2530 Identify the type of conic section whose equation is given and find the vertices and foci.y 2 8y 6x 16 y

2530 Identify the type of conic section whose equation is given and find the vertices and foci.y 2 2y 4x 2 3 4x

2530 Identify the type of conic section whose equation is given and find the vertices and foci.4x 2 4x y 2 0 0,

31 48 Find an equation for the conic that satisfies the given conditions. Parabola, vertex , focus

31 48 Find an equation for the conic that satisfies the given conditions.Parabola, focus , directrix

31 48 Find an equation for the conic that satisfies the given conditions.Parabola, focus , directrix

31 48 Find an equation for the conic that satisfies the given conditions.Parabola, focus , vertex

31 48 Find an equation for the conic that satisfies the given conditions.. Parabola, vertex , vertical axis, passing through

31 48 Find an equation for the conic that satisfies the given conditions.Parabola, horizontal axis, passing through , , and

31 48 Find an equation for the conic that satisfies the given conditions.Ellipse, foci , vertices

31 48 Find an equation for the conic that satisfies the given conditions.Ellipse, foci , vertices

31 48 Find an equation for the conic that satisfies the given conditions.Ellipse, foci , , vertices ,

31 48 Find an equation for the conic that satisfies the given conditions.Ellipse, foci , , vertex

31 48 Find an equation for the conic that satisfies the given conditions.Ellipse, center , vertex , focus

31 48 Find an equation for the conic that satisfies the given conditions.Ellipse, foci , passing through

31 48 Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices , foci

31 48 Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices , foci

31 48 Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices ,

31 48 Find an equation for the conic that satisfies the given conditions.. Hyperbola, vertices

31 48 Find an equation for the conic that satisfies the given conditions.Hyperbola, vertices

31 48 Find an equation for the conic that satisfies the given conditions.Hyperbola, foci , , asymptotes a

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 110 km and apolune altitude 314 km (above the moon). Find an equation of this ellipse if the radius of the moon is 1728 km and the center of the moon is at one focus.

A cross-section of a parabolic reflector is shown in the figure. The bulb is located at the focus and the opening at the focus is 10 cm. (a) Find an equation of the parabola. (b) Find the diameter of the opening , 11 cm from the vertex.

In the LORAN (LOng RAnge Navigation) radio navigation system, two radio stations located at and transmit simul taneous signals to a ship or an aircraft located at . The onboard computer converts the time difference in receiving these signals into a distance difference , and this, according to the definition of a hyperbola, locates the ship or aircraft on one branch of a hyperbola (see the figure). Suppose that station B is located 400 mi due east of station A on a coastline. A ship received the signal from B 1200 micro seconds (s) before it received the signal from A. (a) Assuming that radio signals travel at a speed of 980 ft s, find an equation of the hyperbola on which the ship lies. (b) If the ship is due north of , how far off the coastline is the ship?

Use the definition of a hyperbola to derive Equation 6 for a hyperbola with foci and vertices .

Show that the function defined by the upper branch of the hyperbola is concave upward.

Find an equation for the ellipse with foci and and major axis of length 4.

Determine the type of curve represented by the equation in each of the following cases: (a) , (b) , and (c) . (d) Show that all the curves in parts (a) and (b) have the same foci, no matter what the value of is.

(a) Show that the equation of the tangent line to the parabola at the point can be written as (b) What is the -intercept of this tangent line? Use this fact to draw the tangent line.

Show that the tangent lines to the parabola drawn from any point on the directrix are perpendicular.

. Show that if an ellipse and a hyperbola have the same foci, then their tangent lines at each point of intersection are perpendicular.

Use parametric equations and Simpsons Rule with to estimate the circumference of the ellipse .9x 2 4y 2 36 1.

The planet Pluto travels in an elliptical orbit around the sun (at one focus). The length of the major axis is km and the length of the minor axis is km. Use Simpsons Rule with to estimate the distance traveled by the planet during one complete orbit around the sun.

Find the area of the region enclosed by the hyperbola and the vertical line through a focus.

(a) If an ellipse is rotated about its major axis, find the volume of the resulting solid. (b) If it is rotated about its minor axis, find the resulting volume.

Find the centroid of the region enclosed by the -axis and the top half of the ellipse .9x 2 4y 2 36 x

. (a) Calculate the surface area of the ellipsoid that is generated by rotating an ellipse about its major axis. (b) What is the surface area if the ellipse is rotated about its minor axis?

Let be a point on the ellipse with foci and and let and be the angles between the linesand the ellipse as shown in the figure. Prove that . This explains how whispering galleries and litho tripsy work. Sound coming from one focus is reflected and passes through the other focus. [Hint: Use the formula in Problem 19 on page 271 to show that

Let be a point on the hyperbola with foci and and let and be the angles between the lines , and the hyperbola as shown in the figure. Prove that . (This is the reflection property of thehyperbola. It shows that light aimed at a focus of a hyperbolic mirror is reflected toward the other focus .)