Find the vertices, the foci, and the equations of the

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find the vertex, focus, and directrix of the parabola. Sketch its graph, showing the focus and the directrix.

Find an equation for the parabola shown in the figure.

Find an equation for the parabola shown in the figure.

Find an equation for the parabola shown in the figure.

Find an equation for the parabola shown in the figure.

Find an equation for the parabola shown in the figure.

Find an equation for the parabola shown in the figure.

Find an equation for the parabola shown in the figure.

Find an equation for the parabola shown in the figure.

Find an equation of the parabola that satisfies the given conditions. Focus , directrix

Find an equation of the parabola that satisfies the given conditions. Focus , directrix

Find an equation of the parabola that satisfies the given conditions. Focus , directrix

Find an equation of the parabola that satisfies the given conditions. Focus , directrix

Find an equation of the parabola that satisfies the given conditions. Vertex , directrix

Find an equation of the parabola that satisfies the given conditions. Vertex , directrix

Find an equation of the parabola that satisfies the given conditions. Vertex , directrix

Find an equation of the parabola that satisfies the given conditions. Vertex , directrix

Find an equation of the parabola that satisfies the given conditions. Vertex , focus

Find an equation of the parabola that satisfies the given conditions. Vertex , focus

Find an equation of the parabola that satisfies the given conditions. Vertex , focus

Find an equation of the parabola that satisfies the given conditions. Vertex , focus

Find an equation of the parabola that satisfies the given conditions. Vertex at the origin, symmetric with respect to the y-axis, and passing through the point

Find an equation of the parabola that satisfies the given conditions. Vertex at the origin, symmetric with respect to the y-axis, and passing through the point

Find an equation of the parabola that satisfies the given conditions. Vertex , axis parallel to the x-axis, and passing through the point

Find an equation of the parabola that satisfies the given conditions. Vertex , axis parallel to the x-axis, and y-intercept 1

Find an equation for the set of points in an xy-plane that are equidistant from the point P and the line l.

Find an equation for the set of points in an xy-plane that are equidistant from the point P and the line l.

Find an equation for the set of points in an xy-plane that are equidistant from the point P and the line l.

Find an equation for the set of points in an xy-plane that are equidistant from the point P and the line l.

Find an equation for the indicated half of the parabola. Lower half of

Find an equation for the indicated half of the parabola. Upper half of

Find an equation for the indicated half of the parabola. Right half of

Find an equation for the indicated half of the parabola. Left half of

Find an equation for the indicated half of the parabola. Upper half of

Find an equation for the indicated half of the parabola. Lower half of

Find an equation for the indicated half of the parabola. Left half of

Find an equation for the indicated half of the parabola. Right half of

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. y  x  6  2

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. y   x  3  4

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. x   y  7  3

Determine whether the graph of the equation is the upper, lower, left, or right half of a parabola, and find an equation for the parabola. x  y  4  8

Find an equation for the parabola that has a vertical axis and passes through the given points.

Find an equation for the parabola that has a vertical axis and passes through the given points.

Find an equation for the parabola that has a horizontal axis and passes through the given points.

Find an equation for the parabola that has a horizontal axis and passes through the given points.

Telescope mirror A mirror for a reflecting telescope has the shape of a (finite) paraboloid of diameter 8 inches and depth 1 inch. How far from the center of the mirror will the incoming light collect?

Antenna dish A satellite antenna dish has the shape of a paraboloid that is 10 feet across at the open end and is 3 feet deep. At what distance from the center of the dish should the receiver be placed to receive the greatest intensity of sound waves?

Searchlight reflector A searchlight reflector has the shape of a paraboloid, with the light source at the focus. If the reflector is 3 feet across at the opening and 1 foot deep, where is the focus?

Flashlight mirror A flashlight mirror has the shape of a paraboloid of diameter 4 inches and depth inch, as shown in the figure. Where should the bulb be placed so that the emitted light rays are parallel to the axis of the paraboloid?

Receiving dish A sound receiving dish used at outdoor sporting events is constructed in the shape of a paraboloid, with its focus 5 inches from the vertex. Determine the width of the dish if the depth is to be 2 feet.

Receiving dish Work Exercise 61 if the receiver is 9 inches from the vertex.

Parabolic reflector (a) The focal length of the (finite) paraboloid in the figure is the distance p between its vertex and focus. Express p in terms of r and h. (b) A reflector is to be constructed with a focal length of 10 feet and a depth of 5 feet. Find the radius of the reflector.

Confocal parabolas The parabola has its focus at the origin and axis along the x-axis. By assigning different values to p, we obtain a family of confocal parabolas, as shown in the figure. Such families occur in the study of electricity and magnetism. Show that there are exactly two parabolas in the family that pass through a given point if .

Jodrell Bank radio telescope A radio telescope has the shape of a paraboloid of revolution, with focal length p and diameter of base 2a. From calculus, the surface area S available for collecting radio waves is One of the largest radio telescopes, located in Jodrell Bank, Cheshire, England, has diameter 250 feet and focal length 75 feet. Approximate S to the nearest thousand square feet.

Satellite path A satellite will travel in a parabolic path near a planet if its velocity v in meters per second satisfies the equation , where r is the distance in meters between the satellite and the center of the planet and k is a positive constant. The planet will be located at the focus of the parabola, and the satellite will pass by the planet once. Suppose a satellite is designed to follow a parabolic path and travel within 58,000 miles of Mars, as shown in the figure on the next page. (a) Determine an equation of the form that describes its flight path. (b) For Mars, . Approximate the maximum velocity of the satellite. (c) Find the velocity of the satellite when its y-coordinate is 100,000 miles.

Graph the equation.

Graph the equation.

Graph the parabolas on the same coordinate plane, and estimate the points of intersection.

Graph the parabolas on the same coordinate plane, and estimate the points of intersection.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find the vertices and foci of the ellipse. Sketch its graph, showing the foci.

Find an equation for the ellipse shown in the figure.

Find an equation for the ellipse shown in the figure.

Find an equation for the ellipse shown in the figure.

Find an equation for the ellipse shown in the figure.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find an equation for the ellipse that has its center at the origin and satisfies the given conditions.

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection.

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection.

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection.

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection.

Find an equation for the set of points in an xy-plane such that the sum of the distances from F and is k.

Find an equation for the set of points in an xy-plane such that the sum of the distances from F and is k.

Find an equation for the set of points in an xy-plane such that the sum of the distances from F and is k.

Find an equation for the set of points in an xy-plane such that the sum of the distances from F and is k.

Find an equation for the ellipse with foci F and that passes through P. Sketch the ellipse.

Find an equation for the ellipse with foci F and that passes through P. Sketch the ellipse.

Find an equation for the indicated half of the ellipse. Left half of

Find an equation for the indicated half of the ellipse.Right half of

Find an equation for the indicated half of the ellipse.Upper half of

Find an equation for the indicated half of the ellipse.Lower half of

Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse.

Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse.

Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse.

Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse.

Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse.

Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse.

Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse.

Determine whether the graph of the equation is the upper, lower, left, or right half of an ellipse, and find an equation for the ellipse.

Dimensions of an arch An arch of a bridge is semielliptical, with major axis horizontal. The base of the arch is 30 feet across, and the highest part of the arch is 10 feet above the horizontal roadway, as shown in the figure. Find the height of the arch 6 feet from the center of the base.

Designing a bridge A bridge is to be constructed across a river that is 200 feet wide. The arch of the bridge is to be semielliptical and must be constructed so that a ship less than 50 feet wide and 30 feet high can pass safely through the arch, as shown in the figure on the next page. (a) Find an equation for the arch. (b) Approximate the height of the arch in the middle of the bridge

Earths orbit Assume that the length of the major axis of Earths orbit is 186,000,000 miles and that the eccentricity is 0.017. Approximate, to the nearest 1000 miles, the maximum and minimum distances between Earth and the sun.

Mercurys orbit The planet Mercury travels in an elliptical orbit that has eccentricity 0.206 and major axis of length 0.774 AU. Find the maximum and minimum distances between Mercury and the sun.

Elliptical reflector The basic shape of an elliptical reflector is a hemi-ellipsoid of height h and diameter k, as shown in the figure. Waves emitted from focus F will reflect off the surface into focus . (a) Express the distances and in terms of h and k. (b) An elliptical reflector of height 17 centimeters is to be constructed so that waves emitted from F are reflected to a point that is 32 centimeters from V. Find the diameter of the reflector and the location of F.

Lithotripter operation A lithotripter of height 15 centimeters and diameter 18 centimeters is to be constructed (see the figure). High-energy underwater shock waves will be emitted from the focus F that is closest to the vertex V. (a) Find the distance from V to F. (b) How far from V (in the vertical direction) should a kidney stone be located?

Whispering gallery The ceiling of a whispering gallery has the shape of the hemi-ellipsoid shown in Figure 14, with the highest point of the ceiling 15 feet above the elliptical floor and the vertices of the floor 50 feet apart. If two people are standing at the foci and F, how far from the vertices are their feet?

Oval design An artist plans to create an elliptical design with major axis and minor axis , centered on a door that measures , using the method described by Figure 1. On a vertical line that bisects the door, approximately how far from each end of the door should the pushpins be inserted? How long should the string be?

The planets move around the sun in elliptical orbits. Given the semimajor axis a in millions of kilometers and eccentricity e, graph the orbit for the planet. Center the major axis on the x-axis, and plot the location of the sun at one focus. Earths path ,

The planets move around the sun in elliptical orbits. Given the semimajor axis a in millions of kilometers and eccentricity e, graph the orbit for the planet. Center the major axis on the x-axis, and plot the location of the sun at one focus. Plutos path , e  0.249

Graph the ellipses on the same coordinate plane, and estimate their points of intersection.

Graph the ellipses on the same coordinate plane, and estimate their points of intersection.

Graph the ellipses on the same coordinate plane, and estimate their points of intersection.

Graph the ellipses on the same coordinate plane, and estimate their points of intersection.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci.

Find an equation for the hyperbola shown in the figure.

Find an equation for the hyperbola shown in the figure.

Find an equation for the hyperbola shown in the figure.

Find an equation for the hyperbola shown in the figure.

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci , vertices

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci , vertices

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci , vertices

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci , vertices

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci , conjugate axis of length 4

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci , conjugate axis of length 8

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices , passing through

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices , passing through

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices , asymptotes

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertices , asymptotes

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci , asymptotes

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Foci , asymptotes

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. x-intercepts , asymptotes y  2

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. y-intercepts , asymptotes

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Vertical transverse axis of length 10, conjugate axis of length 14

Find an equation for the hyperbola that has its center at the origin and satisfies the given conditions. Horizontal transverse axis of length 6, conjugate axis of length 2

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. 1 3x  2  y2

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. y2  14 3  x2

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. x2  6x  y2  7

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. x2  4x  4y2  24y 36

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. x2  y2  25

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. x  2x2  y  4

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. 4x2  16x  9y2  36y 16

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. x  4  y2  y

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. x2  3x  3y  6

Identify the graph of the equation as a parabola (with vertical or horizontal axis), circle, ellipse, or hyperbola. 9x2  y2  10  2y

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection.

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection.

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection.

Find the points of intersection of the graphs of the equations. Sketch both graphs on the same coordinate plane, and show the points of intersection.

: Find an equation for the set of points in an xy-plane such that the difference of the distances from F and is k

: Find an equation for the set of points in an xy-plane such that the difference of the distances from F and is k

: Find an equation for the set of points in an xy-plane such that the difference of the distances from F and is k

: Find an equation for the set of points in an xy-plane such that the difference of the distances from F and is k

Find an equation for the hyperbola with foci F and that passes through P. Sketch the hyperbola.

Find an equation for the hyperbola with foci F and that passes through P. Sketch the hyperbola.

Find an equation for the indicated part of the hyperbola.Lower branch of

Find an equation for the indicated part of the hyperbola.Upper branch of

Find an equation for the indicated part of the hyperbola. Left branch of

Find an equation for the indicated part of the hyperbola.Right branch of

Find an equation for the indicated part of the hyperbola. Right halves of the branches of

Find an equation for the indicated part of the hyperbola.Left halves of the branches of

Find an equation for the indicated part of the hyperbola.Upper halves of the branches of

Find an equation for the indicated part of the hyperbola.v

Describe the part of a hyperbola given by the equation.

Describe the part of a hyperbola given by the equation.

Describe the part of a hyperbola given by the equation.

Describe the part of a hyperbola given by the equation.

Describe the part of a hyperbola given by the equation.

Describe the part of a hyperbola given by the equation.

Describe the part of a hyperbola given by the equation.

Describe the part of a hyperbola given by the equation.

The graphs of the equations and are called conjugate hyperbolas. Sketch the graphs of both equations on the same coordinate plane, with and , and describe the relationship between the two graphs.

Find an equation of the hyperbola with foci and vertices , where and

Cooling tower A cooling tower, such as the one shown in the figure, is a hyperbolic structure. Suppose its base diameter is 100 meters and its smallest diameter of 48 meters occurs 84 meters from the base. If the tower is 120 meters high, approximate its diameter at the top.

Airplane maneuver An airplane is flying along the hyperbolic path illustrated in the figure. If an equation of the path is , determine how close the airplane comes to a town located at . (Hint: Let S denote the square of the distance from a point on the path to , and find the minimum value of S.)

Locating a ship A ship is traveling a course that is 100 miles from, and parallel to, a straight shoreline. The ship sends out a distress signal that is received by two Coast Guard stations A and B, located 200 miles apart, as shown in the figure. By measuring the difference in signal reception times, it is determined that the ship is 160 miles closer to B than to A. Where is the ship?

Design of a telescope The Cassegrain telescope design (dating back to 1672) makes use of the reflective properties of both the parabola and the hyperbola. Shown in the figure is a (split) parabolic mirror, with focus at and axis along the line l, and a hyperbolic mirror, with one focus also at and transverse axis along l. Where do incoming light waves parallel to the common axis finally collect?

Graph the hyperbolas on the same coordinate plane, and estimate their first-quadrant point of intersection.

Graph the hyperbolas on the same coordinate plane, and estimate their first-quadrant point of intersection.

Graph the hyperbolas on the same coordinate plane, and determine the number of points of intersection.

Graph the hyperbolas on the same coordinate plane, and determine the number of points of intersection.

Comets path Comets can travel in elliptical, parabolic, or hyperbolic paths around the sun. If a comet travels in a parabolic or hyperbolic path, it will pass by the sun once and never return. Suppose that a comets coordinates in miles can be described by the equation for where the sun is located at a focus, as shown in the figure. (a) Approximate the coordinates of the sun. (b) For the comet to maintain a hyperbolic trajectory, the minimum velocity v of the comet, in meters per second, must satisfy , where r is the distance between the comet and the center of the sun in meters and is a constant. Determine v when r is minimum.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

(a) Describe the graph of a curve C that has the parametrization (b) Change the parametrization to and describe how this changes the graph from part (a). (c) Change the parametrization to and describe how this changes the graph from part (a).

(a) Describe the graph of a curve C that has the parametrization (b) Change the parametrization to and describe how this changes the graph from part (a). (c) Change the parametrization to and describe how this changes the graph from part (a).

Curves , , , and are given parametrically, for t in . Sketch their graphs, and indicate orientations.

Curves , , , and are given parametrically, for t in . Sketch their graphs, and indicate orientations.

The parametric equations specify the position of a moving point P(x, y) at time t. Sketch the graph, and indicate the motion of P as t increases.

The parametric equations specify the position of a moving point P(x, y) at time t. Sketch the graph, and indicate the motion of P as t increases.

Show that ,; are parametric equations of an ellipse with center and axes of lengths 2a and 2b.

Show that ,; and are parametric equations of a hyperbola with center , transverse axis of length 2a, and conjugate axis of length 2b. Determine the values of t for each branch.

(a) Find three parametrizations that give the same graph as the given equation. (b) Find three parametrizations that give only a portion of the graph of the given equation.

(a) Find three parametrizations that give the same graph as the given equation. (b) Find three parametrizations that give only a portion of the graph of the given equation.

Refer to the equations in (1) of Example 6. Find the range and maximum altitude for the given values.

Refer to the equations in (1) of Example 6. Find the range and maximum altitude for the given values.

Refer to the equations in (1) of Example 6. Find the range and maximum altitude for the given values.

Refer to the equations in (1) of Example 6. Find the range and maximum altitude for the given values.

Refer to Example 7. (a) Describe the Lissajous figure given by and for and . (b) Suppose and , where and are positive rational numbers, and write as for positive integers m and n. Show that if , then both and . Conclude that the curve retraces itself every p units of time.

Shown in the figure is the Lissajous figure given by , ; . Find the period of the figurethat is, the length of the smallest t-interval that traces the curve.

Lissajous figures are used in the study of electrical circuits to determine the phase difference  between a known voltage (t)  A sin (  t) and an unknown voltage (t)  B sin (  t   ) having the same frequency. The voltages are graphed parametrically as x  (t) and y  (t). If  is acute, then where is the nonnegative y-intercept and is the maximum y-value on the curve. (a) Graph the parametric curve x  (t) and y  (t) for the specified range of t. (b) Use the graph to approximate  in degrees.

Lissajous figures are used in the study of electrical circuits to determine the phase difference  between a known voltage (t)  A sin (  t) and an unknown voltage (t)  B sin (  t   ) having the same frequency. The voltages are graphed parametrically as x  (t) and y  (t). If  is acute, then where is the nonnegative y-intercept and is the maximum y-value on the curve. (a) Graph the parametric curve x  (t) and y  (t) for the specified range of t. (b) Use the graph to approximate  in degrees.

Lissajous figures are used in the study of electrical circuits to determine the phase difference  between a known voltage (t)  A sin (  t) and an unknown voltage (t)  B sin (  t   ) having the same frequency. The voltages are graphed parametrically as x  (t) and y  (t). If  is acute, then where is the nonnegative y-intercept and is the maximum y-value on the curve. (a) Graph the parametric curve x  (t) and y  (t) for the specified range of t. (b) Use the graph to approximate  in degrees.

Lissajous figures are used in the study of electrical circuits to determine the phase difference  between a known voltage (t)  A sin (  t) and an unknown voltage (t)  B sin (  t   ) having the same frequency. The voltages are graphed parametrically as x  (t) and y  (t). If  is acute, then where is the nonnegative y-intercept and is the maximum y-value on the curve. (a) Graph the parametric curve x  (t) and y  (t) for the specified range of t. (b) Use the graph to approximate  in degrees.

Graph the Lissajous figure in the viewing rectangle [1, 1] by [1, 1] for the specified range of t.

Graph the Lissajous figure in the viewing rectangle [1, 1] by [1, 1] for the specified range of t.

A circle C of radius b rolls on the outside of the circle , and . Let P be a fixed point on C, and let the initial position of P be , as shown in the figure. If the parameter t is the angle from the positive x-axis to the line segment from O to the center of C, show that parametric equations for the curve traced by P (an epicycloid) are

If the circle C of Exercise 53 rolls on the inside of the second circle (see the figure), then the curve traced by P is a hypocycloid. (a) Show that parametric equations for this curve are (b) If , show that , , and sketch the graph.

If in Exercise 53, find parametric equations for the epicycloid and sketch the graph.

The radius of circle B is one-third that of circle A. How many revolutions will circle B make as it rolls around circle A until it reaches its starting point? (Hint: Use Exercise 55.)

Graph the curve.

Graph the curve.

Graph the curve.

Graph the curve.

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure.

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure.

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure.

Graph the given curves on the same coordinate plane, and describe the shape of the resulting figure.

Which polar coordinates represent the same point as ?

Which polar coordinates represent the same point as ?

Change the polar coordinates to rectangular coordinates.

Change the polar coordinates to rectangular coordinates.

Change the polar coordinates to rectangular coordinates.

Change the polar coordinates to rectangular coordinates.

Change the polar coordinates to rectangular coordinates.

Change the polar coordinates to rectangular coordinates.

Change the rectangular coordinates to polar coordinates with and

Change the rectangular coordinates to polar coordinates with and

Change the rectangular coordinates to polar coordinates with and

Change the rectangular coordinates to polar coordinates with and

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Find an equation in x and y that has the same graph as the polar equation. Use it to help sketch the graph in an r  -plane.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

If and are points in an -plane, use the law of cosines to prove that

Prove that the graph of each polar equation is a circle, and find its center and radius.

Refer to Exercise 85 in Section 5.6. Suppose that a radio station has two broadcasting towers located along a north-south line and that the towers are separated by a distance of , where  is the wavelength of the stations broadcasting signal. Then the intensity I of the signal in the direction  can be expressed by the given equation, where is the maximum intensity of the signal. (a) Plot I using polar coordinates with for   [0, 2 ]. (b) Determine the directions in which the radio signal has maximum and minimum intensity.

Refer to Exercise 85 in Section 5.6. Suppose that a radio station has two broadcasting towers located along a north-south line and that the towers are separated by a distance of , where  is the wavelength of the stations broadcasting signal. Then the intensity I of the signal in the direction  can be expressed by the given equation, where is the maximum intensity of the signal. (a) Plot I using polar coordinates with for   [0, 2 ]. (b) Determine the directions in which the radio signal has maximum and minimum intensity.

Graph the polar equation for the indicated values of  , and use the graph to determine symmetries.

Graph the polar equation for the indicated values of  , and use the graph to determine symmetries.

Graph the polar equations on the same coordinate plane, and estimate the points of intersection of the graphs.

Graph the polar equations on the same coordinate plane, and estimate the points of intersection of the graphs.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find the eccentricity, and classify the conic. Sketch the graph, and label the vertices.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find equations in x and y for the polar equations in Exercises 112.

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix.

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix.

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix.

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix.

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix.

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix.

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix.

Find a polar equation of the conic with focus at the pole that has the given eccentricity and equation of directrix.

Find a polar equation of the parabola with focus at the pole and the given vertex.

Find a polar equation of the parabola with focus at the pole and the given vertex.

An ellipse has a focus at the pole with the given center C and vertex V. Find (a) the eccentricity and (b) a polar equation for the ellipse.

An ellipse has a focus at the pole with the given center C and vertex V. Find (a) the eccentricity and (b) a polar equation for the ellipse.

Keplers first law Keplers first law asserts that planets travel in elliptical orbits with the sun at one focus. To find an equation of an orbit, place the pole O at the center of the sun and the polar axis along the major axis of the ellipse (see the figure). (a) Show that an equation of the orbit is where e is the eccentricity and 2a is the length of the major axis. (b) The perihelion distance and aphelion distance are defined as the minimum and maximum distances, respectively, of a planet from the sun. Show that

Keplers first law Refer to Exercise 37. The dwarf planet Pluto travels in an elliptical orbit of eccentricity 0.249. If the perihelion distance is 29.62 AU, find a polar equation for the orbit and estimate the aphelion distance.

Polar equations of conics can be used to describe the motion of comets. These paths can be graphed using the polar equation where e is the eccentricity of the conic and is the perihelion distance measured in AU. (a) For each comet, determine whether its trajectory is elliptical, parabolic, or hyperbolic. (b) The orbit of Saturn has and . Graph both the motion of the comet and the orbit of Saturn in the specified viewing rectangle.

Polar equations of conics can be used to describe the motion of comets. These paths can be graphed using the polar equation where e is the eccentricity of the conic and is the perihelion distance measured in AU. (a) For each comet, determine whether its trajectory is elliptical, parabolic, or hyperbolic. (b) The orbit of Saturn has and . Graph both the motion of the comet and the orbit of Saturn in the specified viewing rectangle.

Polar equations of conics can be used to describe the motion of comets. These paths can be graphed using the polar equation where e is the eccentricity of the conic and is the perihelion distance measured in AU. (a) For each comet, determine whether its trajectory is elliptical, parabolic, or hyperbolic. (b) The orbit of Saturn has and . Graph both the motion of the comet and the orbit of Saturn in the specified viewing rectangle.

Polar equations of conics can be used to describe the motion of comets. These paths can be graphed using the polar equation where e is the eccentricity of the conic and is the perihelion distance measured in AU. (a) For each comet, determine whether its trajectory is elliptical, parabolic, or hyperbolic. (b) The orbit of Saturn has and . Graph both the motion of the comet and the orbit of Saturn in the specified viewing rectangle.

Earths orbit The closest Earth gets to the sun is about 91,405,950 miles, and the farthest Earth gets from the sun is about 94,505,420 miles. Referring to the formulas in Exercise 37, find formulas for a and e in terms of and .

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the vertices and foci of the conic, and sketch its graph.

Find the standard equation of a parabola with a vertical axis that satisfies the given conditions. x-intercepts and , y-intercept 80

Find the standard equation of a parabola with a vertical axis that satisfies the given conditions. x-intercepts and 3, passing through

Find an equation for the conic that satisfies the given conditions. Hyperbola, with vertices and endpoints of conjugate axis

Find an equation for the conic that satisfies the given conditions.Parabola, with focus and directrix

Find an equation for the conic that satisfies the given conditions. Parabola, with focus and directrix

Find an equation for the conic that satisfies the given conditions. Parabola, with vertex at the origin, symmetric to the x-axis, and passing through the poin

Find an equation for the conic that satisfies the given conditions. Ellipse, with vertices and foci

Find an equation for the conic that satisfies the given conditions. Hyperbola, with foci and vertices V 5, 0

Find an equation for the conic that satisfies the given conditions. Hyperbola, with vertices and asymptotes

Find an equation for the conic that satisfies the given conditions. Ellipse, with foci and passing through the point

Find an equation for the conic that satisfies the given conditions. Ellipse, with eccentricity and endpoints of minor axis

Find an equation for the conic that satisfies the given conditions. Ellipse, with eccentricity and foci

Find an equation for the portion of the conic.Right half of

Find an equation for the portion of the conic.Lower half of

Find an equation for the portion of the conic. Left half of

Find an equation for the portion of the conic.Upper half of

Find an equation for the portion of the conic. Right branch of

Find an equation for the portion of the conic.Lower branch of

(a) Determine A so that the point is on the conic . (b) Is the conic an ellipse or a hyperbola?

If a square with sides parallel to the coordinate axes is inscribed in the ellipse , express the area A of the square in terms of a and b.

Find the standard equation of the circle that has center at the focus of the parabola and passes through the origin.

Focal length and angular velocity A cylindrical container, partially filled with mercury, is rotated about its axis so that the angular speed of each cross section is . From physics, the function f, whose graph generates the inside surface of the mercury (see the figure), is given by where k is a constant. Determine the angular speed that will result in a focal length of 2 feet.

An ellipse has a vertex at the origin and foci and , as shown in the figure. If the focus at is fixed and is on the ellipse, show that approaches 4px as . (Thus, as , the ellipse takes on the shape of a parabola.)

Alpha particles In 1911, the physicist Ernest Rutherford (18711937) discovered that if alpha particles are shot toward the nucleus of an atom, they are eventually repulsed away from the nucleus along hyperbolic paths. The figure illustrates the path of a particle that starts toward the origin along the line and comes within 3 units of the nucleus. Find an equation of the path.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Find an equation in x and y whose graph contains the points on the curve C. Sketch the graph of C, and indicate the orientation.

Curves , , , and are given parametrically for t in . Sketch their graphs, and discuss their similarities and differences. :, :, :, :,

Refer to the equations in (1) of Example 6 in Section 10.4. Find the range and maximum altitude for , , and .

List two polar coordinate points that represent the same point as .

Change to rectangular coordinates.

Change to polar coordinates with and .

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find a polar equation that has the same graph as the equation in x and y.

Find an equation in x and y that has the same graph as the polar equation.

Find an equation in x and y that has the same graph as the polar equation.

Find an equation in x and y that has the same graph as the polar equation.

Find an equation in x and y that has the same graph as the polar equation.

Find an equation in x and y that has the same graph as the polar equation.

Find an equation in x and y that has the same graph as the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

Sketch the graph of the polar equation.

On a parabola, the line segment through the focus, perpendicular to the axis, and intercepted by the parabola is called the focal chord or latus rectum. The length of the focal chord is called the focal width. Find a formula for the focal width w in terms of the focal length .

On the graph of a hyperbola with center at the origin O, draw a circle with center at the origin and radius , where F denotes a focus of the hyperbola. What relationship do you observe

A point is on an ellipse if and only if If , derive the general equation of an ellipse that is,

A point is on a hyperbola if and only if If , derive the general equation of a hyperbola that is,

A point is the same distance from as it is from the circle , as illustrated in the figure. Show that the collection of all such points forms a branch of a hyperbola, and sketch its graph.

Design of a telescope Refer to Exercise 78 in Section 10.3. Suppose the upper branch of the hyperbola (shown) has equation and an equation of the parabola is . Find d in terms of a and b.

Maximizing a projectiles range As in Example 6 in Section 10.4, suppose a projectile is to be fired at a speed of 1024 from a height of 2304 feet. Approximate the angle that maximizes the range.

Generalizations for a projectiles path If , the equations in (1) of Example 6 in Section 10.4 become ,; . Show that each statement is true. (a) The projectile strikes the ground when (b) The range r of the projectile is (c) The angle that maximizes the range r is 45. (d) The path of the projectile in rectangular coordinates is (e) The time at which the maximum height is reached is (f) The maximum height reached is

Investigating a Lissajous figure Find an equation in x and y for the curve from Example 7 in Section 10.4 given by ,;

Sketch the graphs of the equations , , and for . Try as many values of as necessary to generalize results concerning the graphs of and , where .

Generalized roses Examine the graph of for odd values of n and even values of n. Derive an expression for the leaf angle (the number of degrees between consecutive pole values). What other generalizations do you observe? How do the graphs change if sin is replaced by cos?

Figure 17 of Section 10.5 shows the circles and . Solve this system of equations for solutions. Now find equations in x and y that have the same graphs as the polar equations. Solve this system for solutions, convert them to solutions, and explain why your answer to the first system did not reveal the solution at the pole.

Sketch the graph of the parabola . Label its vertex, focus, and directrix.

Find an equation for the parabola shown in the figure.

Find an equation of the parabola that has vertex and the x-axis as its directrix.

Find an equation for the lower half of the parabola .

Find an equation for the parabola that has a horizontal axis and passes through the points , , and .

A searchlight reflector has the shape of a paraboloid, with the light source at the focus. If the reflector is 40 inches across at the opening and 7 inches deep, where is the focus?

Sketch the graph of the ellipse . Label its vertices, foci, and endpoints of the minor axis.

Find an equation for the ellipse that has its center at the origin, vertices , and foci .

An ellipse has vertices and minor axis length 8. Find its eccentricity.

Find an equation for the set of points in an xy-plane such that the sum of the distances from and is .

Find an equation for the left half of the ellipse .

An arch of a bridge is semi-elliptical, with major axis horizontal. The base of the arch is 40 feet across, and the highest part of the arch is 12 feet above the horizontal roadway. Approximate the height, to one decimal place, of the arch 8 feet from the center of the base.

Sketch the graph of the hyperbola . Label its vertices, foci, and asymptotes.

Given that a hyperbola has vertices and foci , find equations of the asymptotes of the hyperbola.

Find an equation for the hyperbola that has its center at the origin, foci , and conjugate axis of length 6.

Find an equation for the left branch of the hyperbola .

Find an equation in x and y whose graph contains the points on the curve of C, where C has the parametrization . Describe the graph of C, including its orientation.

Find an equation in x and y whose graph contains the points on the curve of C, where C has the parametrization Sketch the graph of C, and indicate the orientation.

Write a parametrization for the bottom branch of .

Describe the graph (and orientation for increasing values of t) of a curve C that has the parametrization .

Suppose a projectile is fired at a speed of 544 ftsec at an angle of 30 from the horizontal from a height of 960 feet. Find the range of the projectile.

Change the rectangular coordinates to polar coordinates with and .

Find a polar equation that has the same graph as .

Find an equation in x and y that has the same graph as .

Sketch the graph of .

Sketch the graph of .

Find the eccentricity of . Classify the conic, sketch the graph, and label the vertices.

Find an equation in x and y for .

Find a polar equation of the conic with focus at the pole that has eccentricity and directrix .