Radio towers A and 8 , 200 kilomete rs apart. are situated along the coast, wilh A

In Exercises 1-4, find the vertkes and locate the foci of each hyperbola hliJh the given equation. 71zen mal!'h e.ach equation to one of the graphs that are shown and labeled (a)-(d). t2 )''2 x l y2 1. 7 - , - 1 a. y b. y 4 -3 -4 c. d.

In Exercises 1-4, find the vertkes and locate the foci of each hyperbola hliJh the given equation. 71zen mal!'h e.ach equation to one of the graphs that are shown and labeled (a)-(d). x l y2 1. 7 - , - 1 2. , - 4 - 1 a. y b. y 4 -3 -4 c. d.

In Exercises 1-4, find the vertkes and locate the foci of each hyperbola hliJh the given equation. 71zen mal!'h e.ach equation to one of the graphs that are shown and labeled (a)-(d). x2 y2 4 - 1 a. y b. y 4 -3 -4 c. d.

In Exercises 1-4, find the vertkes and locate the foci of each hyperbola hliJh the given equation. 71zen mal!'h e.ach equation to one of the graphs that are shown and labeled (a)-(d). >'z xl 4. , - 4 - 1 a. y b. y 4 -3 -4 c. d.

In Exercises 5-12, find the standard form of the equation of each hyperbola satisfying the given conditions. f'<lci: (0, - 3), (0, 3): vertices: (0, - I), (0, I)

In Exercises 5-12, find the standard form of the equation of each hyperbola satisfying the given conditions. f'<lci: (0, - 6), (0, 6); vcrtiocs: (0, - 2), (0, 2)

In Exercises 5-12, find the standard form of the equation of each hyperbola satisfying the given conditions. f'<lci: (- 4, 0), (4, 0); vertiocs: (- 3, 0), (3, 0)

In Exercises 5-12, find the standard form of the equation of each hyperbola satisfying the given conditions. f'<lci: (- 7, 0), (7, 0); vertioes: (- 5, 0), (5, 0)

In Exercises 5-12, find the standard form of the equation of each hyperbola satisfying the given conditions. Endpoints of transverse a.,is: (0, - 6), (0. 6); asymptote: y - 2t

In Exercises 5-12, find the standard form of the equation of each hyperbola satisfying the given conditions. Endpoints of transverse axis: (- 4, 0), (4, 0): asymptote: y - 2t

In Exercises 5-12, find the standard form of the equation of each hyperbola satisfying the given conditions. Center: (4. - 2): Focus: (7. - 2); vertex: (6. - 2)

In Exercises 5-12, find the standard form of the equation of each hyperbola satisfying the given conditions. Center: (- 2. I); Focus: (- 2, 6); vertex: (- 2. 4)

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. x2 )'2 r2 yl 25 - 1

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. r2 yl ;6 - 25

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. Oo' - 64 - I

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 2 y2 - I 16. i44 - St - I

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. yl xl yl xz i6 - J6' - I

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. yl xz - I 25 - 64 - I

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 4y 2 - x2 - 1

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 9y2 - x2 - l

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 9x2 - 4y2 - 36

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 4x2 - 25y2 - 100

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 9y' - 25x2 - 225

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. 16/ - 9x2 - 144

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. y - =Vx2 - 2

In Exercises 13-26, use vertices and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. y - Vx2 - 3

In Exercises 27-32, find the standard form of the equation of each II yperbo/a. y

In Exercises 27-32, find the standard form of the equation of each II yperbo/a. y

In Exercises 27-32, find the standard form of the equation of each II yperbo/a. y

In Exercises 27-32, find the standard form of the equation of each II yperbo/a. y

In Exercises 27-32, find the standard form of the equation of each II yperbo/a. y

In Exercises 27-32, find the standard form of the equation of each II yperbo/a. y

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (x + 4}' 9 (y + 3)' 16

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (x + 2}' 9 (y - 1 )' 25 - 1

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (x + 3}' y' - 25 u; - l

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (x + 2}' 9 ;;.... '- 1 25

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (y + 2)' (.t - I)' . 4 16

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (v - 2}' (x + 1)2 36 49 - 1

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (x - 3)1 - 4(y + 3)2 - 4

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (x + 3)1 - 9(y - 4}' - 9

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. x - I )1 - (y - 2)' - 3

In Exercises 33-42, use the center. venices. and asymptotes to graph each hyperbola. L<>cate the foci and find the equations of the asymptoli!s. (y - 2)1 - (x + 3)2 - 5

In Exercises 43-50, convert each equation to standard form by completing the squart! on x andy. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. \ 2 - y' - 2< - 4y - 4 - 0

In Exercises 43-50, convert each equation to standard form by completing the squart! on x andy. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. 4x1 - y2 + 32< + 6y + 39 - 0

In Exercises 43-50, convert each equation to standard form by completing the squart! on x andy. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. 16x1 - l + 64x - 2y + 67 - 0

In Exercises 43-50, convert each equation to standard form by completing the squart! on x andy. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. 9y' - 4x1 - 18y + 24x - 63 - 0

In Exercises 43-50, convert each equation to standard form by completing the squart! on x andy. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. 4x1 - 9y2 - 16x + 54y - 101 - 0

In Exercises 43-50, convert each equation to standard form by completing the squart! on x andy. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. 4x1 - 9y1 + 8x - 18y - 6 - 0

In Exercises 43-50, convert each equation to standard form by completing the squart! on x andy. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. 4x1 - 25y' - 32x + 164 - 0

In Exercises 43-50, convert each equation to standard form by completing the squart! on x andy. Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. 9x2 - 16y1 - 36x - 64y + 116 - 0

In Exercises 51-56, graph each relation. Use the relation's graph to determine its domain and range. x2 y2 51. - - - - 1 9 16

In Exercises 51-56, graph each relation. Use the relation's graph to determine its domain and range. x2 y2 25 - 4 - 1

In Exercises 51-56, graph each relation. Use the relation's graph to determine its domain and range. x2 y2 9 +u;

In Exercises 51-56, graph each relation. Use the relation's graph to determine its domain and range. x2 y2 24 + 4 -1

In Exercises 51-56, graph each relation. Use the relation's graph to determine its domain and range. v' x' ss. 16 9

In Exercises 51-56, graph each relation. Use the relation's graph to determine its domain and range. y2 x2 4 - 25 - 1

In Exercises 57-60, find the solution set for each system by graphing both of tiJe system's equations in the same rectangular coordinate system and finding points of intersection. Check all solufions in boih equation'i. x 2 - y' - 4 x1 +y1 - 4

In Exercises 57-60, find the solution set for each system by graphing both of tiJe system's equations in the same rectangular coordinate system and finding points of intersection. Check all solufions in boih equation'i. x2 - y2 - 9 ? 2 9 x- + y

In Exercises 57-60, find the solution set for each system by graphing both of tiJe system's equations in the same rectangular coordinate system and finding points of intersection. Check all solufions in boih equation'i. 9x1 + y2 - 9 y2 - 9x2 - 9

In Exercises 57-60, find the solution set for each system by graphing both of tiJe system's equations in the same rectangular coordinate system and finding points of intersection. Check all solufions in boih equation'i. 4x2 + y2 - 4 y' - 4x1 - 4

An explosion is recorded by two microphones that are 1 mile apart. Microphone M 1 received the sound 2 seconds before microphone M2. Assuming sound travels at llOO feet per second, detennine the possible locations of the explosion rc.lative to the location of the microphones.

Radio towers A and 8 , 200 kilomete rs apart. are situated along the coast, wilh A located due west of B. Simultaneous radio signals a re sent from each tower to a ship, with the signal from B received 500 microseconds be fore the signal from A. a. Assuming that the radio signals travel 300 meters per microsecond, determine the equation of the hyperbola on which the ship is located. b. If the ship lies due north of tower B, how far out a t sea is it?

An architect designs two houses that are shaped a nd positioned like a pa rt or the branches o f the hype rbola whose equation is 625y2 - 400x2 - 250.000, where x a nd y are in yards. How far apart are the houses a t their closest point'/

Scattering experiments, in which moving pa rticles are deHected by various forces, led to the concept of the nucleus of an atom. In 1911, the physicist Ernest Rutherford (1871 - 1937) discovered that when a lpha particles a rc directed toward the nuclei of gold atoms, they are eventually deflected along hyperbolic paths, illustrated in t he figure. If a particle gets as close as 3 units to the nucleus along a hype rbolic path with an asymptote given by y - , what is the equation of its path? y

Moire patterns. such as those shown in Exercises 65-66, can appear when two repetitive pall ems overlap to produce a third, sometimes unintended, patient 65. a. In each exercise. use the name of a conic section to describe the moiri pattern. b. Se{e(:t one of the following equations that can possibly describe a conic :u!ction within the moire pallern: x2 + yl - I; x2 - yl - 1; x2 + 4y2 - 4.

Moire patterns. such as those shown in Exercises 65-66, can appear when two repetitive pall ems overlap to produce a third, sometimes unintended, patient 65. a. In each exercise. use the name of a conic section to describe the moiri pattern. b. Se{e(:t one of the following equations that can possibly describe a conic :u!ction within the moire pallern: x2 + yl - I; x2 - yl - 1; x2 + 4y2 - 4.

What is a hyperbola?

Describe how to graph 9 - T - 1.

Describe how to locate the foci of the graph of 9 - T - I.

Describe one similarity and one difference between the x' y' y' x' graphs of- 1 and - - - - I . 9 I 9 I

Describe one similarity and one difference between the x2 y' ~t - 3)2 (y + 3)2 graphs of 9 - T - 1 and 9 - 1 - 1

How can you distinguish an ellipse from a hyperbola by looking at their equations?

In 1992, a NASA team began a project called Spaceguard Survey, caJiing for an international watch for comets that might collide with Earth. Why is it more difficult to detect a possible "doomsday cometwith a hyperbolic orbit than one with an elliptical orbit?

Use a graphing utility to graph any five of the hyperbolas that you graphed by hand in Exercises 13-26.

Use a graphing utility to graph any three of the hyperbolas that you graphed by hand in Exercises 33-42. First solve the gi\'en equation for y by using the. square rool property. Enter eac.h of the two resulting equations to produce each branch of the hyperbola.

Use a graphing utility to graph any one of the hyperbolas that you graphed by hand in Exercises 43-50. Write the equation as a quadratic equation in y and use the quadratic formula to solve for y. Enter each of the two resulting equations to produoe each branch of the hyperbola.

Use a graphing utility to graph: - ~- - 0. Is the graph a \"2 y2 hyperbola? In general, what is the graph of 2 - b' - 0'/

Graph - - - - I and '-- - - -I in the same viewing al b2 al bl rectangle for values of a1 and b2 of your choice. Describe the relationship between the two graphs.

Write 4x2 - 6xy + 2y' - 3x + tOy - 6 - 0 as a quadratic equation in y and then use the quadratic fom1ula to express y in tenns of x. Graph the resulting two equations using a graphing utility in a [- 50, 70, 10] by (- 30, 50, 10] ,;e\\1ng rectangle. What effect does the xy-term have o n the graph of the resulting hyperbola? What problems would you encounter if you attempted to write the given equation in standard form by completing the square?

Graph i6 - 9 - 1 and 1'6 - 9 - I m the same viewing rectangle. Explain why the graphs are nol the same.

In Exercises 81-84, determine whether each statement makes sense or does not make sense, and explain your reasoning. I changed the addition in an ellipse's equation to subtraction and this changed its elongation from horizontal to vertical.

In Exercises 81-84, determine whether each statement makes sense or does not make sense, and explaitn your reasoning. I noticed that the definition of a hyperbola closely resembles that of an ellipse in that it depends on the distances between a set of points in a plane to two fixed points, the foci.

In Exercises 81-84, determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed a hyperbola centered at the origin that had y-intercepts but no x-intercepts.

In Exercises 81-84, determine whether each statement makes sense or does not make sense, and explaitJ your reasoning. I graphed a hyperbola centered at the origin that was symmetric with respect to the t-axis and also symmetric with respect to they-axis.

In Exercises 85-88, determine whether each statement is true or false. If the statemelll is false. make the necessary change(s) to produce a true statement. If one branch of a hyperbola is removed from a graph. then the branch that remains must define y as a function of x.

In Exercises 85-88, determine whether each statement is true or false. If the statemelll is false. make the necessary change(s) to produce a true statement. All points on the asymptotes of a hyperbola also satisfy the hyperbola's equation.

In Exercises 85-88, determine whether each statement is true or false. If the statemelll is false. make the necessary change(s) to produce a true statement. 111e graph 2 y - . , x2 yl of - - :.- - 1 does not intersec.t the line

In Exercises 85-88, determine whether each statement is true or false. If the statemelll is false. make the necessary change(s) to produce a true statement. Two differe.nt hyperbolas can never share the same asymptotes.

What happens to the shape of the graph of a' - b' - 1 as - .:o where c1 - a1 + b2 ?

Find the standard form of the equation of the hyperbola "1th vertices (5, - 6) and (5. 6), passing through (0, 9).

Find the equation of a hyperbola whose asymptotes are perpendicular.

Exercises 92-94 will help you prepare for the material covered in the next section. In Exerdses 92-93, graph each parabola with the gil'en equation. y - x2 +4x - 5

Exercises 92-94 will help you prepare for the material covered in the next section. In Exerdses 92-93, graph each parabola with the gil'en equation. y - - 3(x - If + 2

Exercises 92-94 will help you prepare for the material covered in the next section. Isolate the terms involving yon the left side of the equation: y2 + 2y + 12x - 23 - 0. Then write the equation in an equivalent fom1 by completing the square on the left side.