Guided Practice. Application. Practice and Apply. Homework Help. Extra Practice.

Transcription

1 Circles Vocabular circle center tangent Write equations of circles. Graph circles. are circles important in air traffic control? Radar equipment can be used to detect and locate objects that are too far awa to be seen b the human ee. The radar sstems at major airports can tpicall detect and track aircraft up to 5 to 70 miles in an direction from the airport. The boundar of the region that a radar sstem can monitor can be modeled b a circle. EQUATINS F CIRCLES A circle is the set of all points in a plane that are equidistant from a given point in the plane, called the center. An segment whose endpoints are the center and a point on the circle is a radius of the circle. Assume that (, ) are the coordinates of a point on the circle at the right. The center is at (h, k), and the radius is r. You can find an equation of the circle b using the Distance Formula. ( 1 ) ( 1 ) d ( ) h ( k) r Distance Formula ( 1, 1 ) (h, k), (, ) (, ), d r ( h) ( k) r Square each side. radius (, ) r (h, k) center Equation of a Circle The equation of a circle with center (h, k) and radius r units is ( h) ( k) r. Eample 1 Write an Equation Given the Center and Radius NUCLEAR PWER In 1986, a nuclear reactor eploded at a power plant about 110 kilometers north and 15 kilometers west of Kiev. At first, officials evacuated people within 30 kilometers of the power plant. Write an equation to represent the boundar of the evacuated region if the origin of the coordinate sstem is at Kiev. Since Kiev is at (0, 0), the power plant is at ( 15, 110). The boundar of the evacuated region is the circle centered at ( 15, 110) with radius 30 kilometers. ( h) ( k) r Equation of a circle [ ( 15)] ( 110) 30 (h, k) ( 15, 110), r 30 ( 15) ( 110) 900 Simplif. The equation is ( 15) ( 110) Chapter 8 Conic Sections

2 Eample Write an Equation Given a Diameter Write an equation for a circle if the endpoints of a diameter are at (5, ) and (, 6). Eplore Plan Solve To write an equation of a circle, ou must know the center and the radius. You can find the center of the circle b finding the midpoint of the diameter. Then ou can find the radius of the circle b finding the distance from the center to one of the given points. First find the center of the circle. (h, k) 1, 1 5 ( ), ( 6) 3, 3, 1 Midpoint Formula ( 1, 1 ) (5, ), (, ) (, 6) Add. Simplif. Now find the radius. r ( 1 ) ( 1 ) Distance Formula 3 5 ( 1 ) ( 5) 7 ( 1, 1 ) (5, ), (, ) 3, 1 Subtract. 1 9 Simplif. The radius of the circle is 1 9 units, so r 1 9. Substitute h, k, and r into the standard form of the equation of a circle. An equation of the circle is 3 [ ( 1)] 1 9 or 3 ( 1) 1 9. Eamine Each of the given points satisfies the equation, so the equation is reasonable. A line in the plane of a circle can intersect the circle in zero, one, or two points. A line that intersects the circle in eactl one point is said to be tangent to the circle. The line and the circle are tangent to each other at this point. Eample 3 Write an Equation Given the Center and a Tangent Write an equation for a circle with center at (, 3) that is tangent to the -ais. Sketch the circle. Since the circle is tangent to the -ais, its radius is 3. An equation of the circle is ( ) ( 3) 9. (, 3) Lesson 8-3 Circles 7

3 GRAPH CIRCLES You can use completing the square, smmetr, and transformations to help ou graph circles. The equation ( h) ( k) r is obtained from the equation r b replacing with h and with k. So, the graph of ( h) ( k) r is the graph of r translated h units to the right and k units up. Eample Graph an Equation in Standard Form Find the center and radius of the circle with equation 5. Then graph the circle. The center of the circle is at (0, 0), and the radius is 5. The table lists some integer values for and that satisf the equation. Since the circle is centered at the origin, it is smmetric about the -ais. Therefore, the points at ( 3, ), (, 3) and ( 5, 0) lie on the graph The circle is also smmetric about the -ais, so the points at (, 3), ( 3, ), (0, 5), (3, ), and (, 3) lie on the graph. Graph all of these points and draw the circle that passes through them. 5 Eample 5 Graph an Equation Not in Standard Form Find the center and radius of the circle with equation Then graph the circle. Complete the squares ( ) ( ) 5 The center of the circle is at (, ), and the radius is 5. In the equation from Eample, has been replaced b, and has been replaced b. The graph is the graph from Eample translated units to the right and down units Concept Check 1. PEN ENDED Write an equation for a circle with center at (6, ). 8 Chapter 8 Conic Sections. Write in standard form b completing the square. Describe the transformation that can be applied to the graph of 6 to obtain the graph of the given equation. 3. FIND THE ERRR Juwan sas that the circle with equation ( ) 36 has radius 36 units. Luc sas that the radius is 6 units. Who is correct? Eplain our reasoning.

4 Guided Practice. Write an equation for the graph at the right. GUIDED PRACTICE KEY Write an equation for the circle that satisfies each set of conditions. 5. center ( 1, 5), radius units 6. endpoints of a diameter at (, 1) and (, 5) 7. center (3, 7), tangent to the -ais Find the center and radius of the circle with the given equation. Then graph the circle. 8. ( ) ( 1) 9 9. ( 1) ( ) Application AERSPACE For Eercises 1 and 15, use the following information. In order for a satellite to remain in a circular orbit above the same spot on Earth, the satellite must be 35,800 kilometers above the equator. 1. Write an equation for the orbit of the satellite. Use the center of Earth as the origin and 600 kilometers for the radius of Earth. 15. Draw a labeled sketch of Earth and the orbit to scale. Practice and Appl Homework Help For Eercises See Eamples , 5 Write an equation for each graph Etra Practice See page 85. The epicenter of an earthquake can be located b using the equation of a circle. Visit webquest to continue work on our WebQuest project. Write an equation for the circle that satisfies each set of conditions. 18. center (0, 3), radius 7 units 19. center ( 8, 7), radius 1 unit 0. endpoints of a diameter at ( 5, ) and (3, 6) 1. endpoints of a diameter at (11, 18) and ( 13, 19). center (8, 9), passes through (1, ) 3. center 13,, passes through the origin. center at ( 8, 7), tangent to -ais 5. center at (, ), tangent to -ais 6. center in the first quadrant; tangent to 3, 5, and the -ais 7. center in the second quadrant; tangent to 1, 9, and the -ais Lesson 8-3 Circles 9

5 8. LANDSCAPING The design of a garden is shown at the right. A pond is to be built in the center region. What is the equation of the largest circular pond centered at the origin that would fit within the walkwas? 9. EARTHQUAKES The Universit of Southern California is located about.5 miles west and about.8 miles south of downtown Los Angeles. Suppose an earthquake occurs with its epicenter about 0 miles from the universit. Assume that the origin of a coordinate plane is located at the center of downtown Los Angeles. Write an equation for the set of points that could be the epicenter of the earthquake. Find the center and radius of the circle with the given equation. Then graph the circle. Earthquakes Southern California has about 10,000 earthquakes per ear man of which occur at or near the San Andreas fault. Most are too small to be felt. Source: ( ) ( 3) ( 1) ( 3) ( 7) ( 3) ( 3) ( 7) 冢 兹5苶 冣 冢 兹3苶 冣 RADI The diagram at the right shows the relative locations of some cities in North Dakota. The -ais represents Interstate 9. The scale is 1 unit 30 miles. While driving west on the highwa, Doralina is listening to a radio station in Minot. She estimates the range of the signal to be 10 miles. How far west of Bismarck will she be able to pick up the signal? Minot Fargo Bismarck 9. CRITICAL THINKING A circle has its center on the line with equation. The circle passes through (1, 3) and has a radius of 兹5苶 units. Write an equation of the circle. 50. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. Wh are circles important in air traffic control? Include the following in our answer: an equation of the circle that determines the boundar of the region where planes can be detected if the range of the radar is 50 miles and the radar is at the origin, and how an air traffic controller s job would be different for a region whose boundar is modeled b 900 instead of Chapter 8 Conic Sections

6 Standardized Test Practice Graphing Calculator 51. Find the radius of the circle with equation A B C 8 D 8 5. Find the center of the circle with equation A ( 10, 6) B (1, 1) C (10, 6) D (5, 3) CIRCLES For Eercises 53 56, use the following information. Since a circle is not the graph of a function, ou cannot enter its equation directl into a graphing calculator. Instead, ou must solve the equation for. The result will contain a smbol, so ou will have two functions. 53. Solve ( 3) 16 for. 5. What two functions should ou enter to graph the given equation? 55. Graph ( 3) 16 on a graphing calculator. 56. Solve ( 3) 16 for. What parts of the circle do the two epressions for represent? Maintain Your Skills Mied Review Identif the coordinates of the verte and focus, the equations of the ais of smmetr and directri, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. (Lesson 8-) ( 3) 59. Find the midpoint of the line segment with endpoints at the given coordinates. (Lesson 8-1) 60. (5, 7), (3, 1) 61. (, 9), (, 5) 6. (8, 0), ( 5, 1) Find all of the rational zeros for each function. (Lesson 7-5) 63. f() g() Getting Read for the Net Lesson 65. PHTGRAPHY The perimeter of a rectangular picture is 86 inches. Twice the width eceeds the length b inches. What are the dimensions of the picture? (Lesson 3-) PREREQUISITE SKILL Solve each equation. Assume that all variables are positive. (To review solving quadratic equations, see Lesson 6-.) 66. c c a a b b P ractice Quiz 1 Lessons 8-1 through 8-3 Find the distance between each pair of points with the given coordinates. (Lesson 8-1) 1. (9, 5), (, 7). (0, 5), (10, 3) Identif the coordinates of the verte and focus, the equations of the ais of smmetr and directri, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. (Lesson 8-) Find the center and radius of the circle with equation ( ) 9. Then graph the circle. (Lesson 8-3) Lesson 8-3 Circles 31

7 A Preview of Lesson 8- Investigating Ellipses Follow the steps below to construct another tpe of conic section. Step 1 Place two thumbtacks in a piece of cardboard, about 1 foot apart. Step Tie a knot in a piece of string and loop it around the thumbtacks. Step 3 Place our pencil in the string. Keep the string tight and draw a curve. Step Continue drawing until ou return to our starting point. The curve ou have drawn is called an ellipse. The points where the thumbtacks are located are called the foci of the ellipse. Foci is the plural of focus. Model and Analze Place a large piece of grid paper on a piece of cardboard. 1. Place the thumbtacks at (8, 0) and ( 8, 0). Choose a string long enough to loop around both thumbtacks. Draw an ellipse.. Repeat Eercise 1, but place the thumbtacks at (5, 0) and ( 5, 0). Use the same loop of string and draw an ellipse. How does this ellipse compare to the one in Eercise 1? ( 8, 0) ( 5, 0) (5, 0) (8, 0) Place the thumbtacks at each set of points and draw an ellipse. You ma change the length of the loop of string if ou like. 3. (1, 0), ( 1, 0). (, 0), (, 0) 5. (1, ), ( 10, ) Make a Conjecture In Eercises 6 10, describe what happens to the shape of an ellipse when each change is made. 6. The thumbtacks are moved closer together. 7. The thumbtacks are moved farther apart. 8. The length of the loop of string is increased. 9. The thumbtacks are arranged verticall. 10. ne thumbtack is removed, and the string is looped around the remaining thumbtack. 11. Pick a point on one of the ellipses ou have drawn. Use a ruler to measure the distances from that point to the points where the thumbtacks were located. Add the distances. Repeat for other points on the same ellipse. What relationship do ou notice? 1. Could this activit be done with a rubber band instead of a piece of string? Eplain. 3 Chapter 8 Conic Sections

8 Ellipses Vocabular ellipse foci major ais minor ais center Write equations of ellipses. Graph ellipses. are ellipses important in the stud of the solar sstem? Fascination with the sk has caused people to wonder, observe, and make conjectures about the planets since the beginning of histor. Since the earl 1600s, the orbits of the planets have been known to be ellipses with the Sun at a focus. EQUATINS F ELLIPSES As ou discovered in the Algebra Activit on page 3, an ellipse is the set of all points in a plane such that the sum of the distances from two fied points is constant. The two fied points are called the foci of the ellipse. The ellipse at the right has foci at (5, 0) (, ) and ( 5, 0). The distances from either of the -intercepts to the foci are units and 1 units, so the sum of the distances from an point with coordinates (, ) on the ellipse to the foci is 1 units. You can use the Distance Formula and the definition of an ellipse to find an equation of this ellipse. ( 5, 0) (5, 0) The distance between the distance between (, ) and ( 5, 0) (, ) and (5, 0) 1. ( ) 5 ( ) 5 1 ( ) 5 1 ( ) 5 Isolate the radicals. ( 5) ( ) 5 ( 5) ( ) ( ) 5 Simplif ( ) 5 Divide each side b [( 5) ] An equation for this ellipse is 1. 9 Square each side. Distributive Propert Simplif. Square each side. Divide each side b Lesson 8- Ellipses 33

9 TEACHING TIP Stud Tip Vertices of Ellipses The endpoints of each ais are called the vertices of the ellipse. Ever ellipse has two aes of smmetr. The points at which the ellipse intersects its aes of smmetr determine two segments with endpoints on the ellipse called the major ais and the minor ais. The aes intersect at the center of the ellipse. The foci of an ellipse alwas lie on the major ais. Stud the ellipse at the right. The sum of the distances from the foci to an point on the Major ais ellipse is the same as the length of the major ( a, 0) ais, or a units. The distance from the center to a a (a, 0) either focus is c units. B the Pthagorean b Theorem, a, b, and c are related b the equation c a b c. Notice that the - and -intercepts, F 1 ( c, 0) F ( a, 0) and (0, b), satisf the quadratic (c, 0) equation a b 1. This is the standard form of the equation of an ellipse with its center at the origin and a horizontal major ais. Center Minor ais Equations of Ellipses with Centers at the rigin Standard Form of Equation a b 1 a b 1 Direction of Major Ais horizontal vertical Foci (c, 0), ( c, 0) (0, c), (0, c) Length of Major Ais a units a units Length of Minor Ais b units b units In either case, a b and c a b. You can determine if the foci are on the -ais or the -ais b looking at the equation. If the term has the greater denominator, the foci are on the -ais. If the term has the greater denominator, the foci are on the -ais. Eample 1 Write an Equation for a Graph Write an equation for the ellipse shown at the right. In order to write the equation for the ellipse, we need to find the values of a and b for the ellipse. We know that the length of the major ais of an ellipse is a units. In this ellipse, the length of the major ais is the distance between the points at (0, 6) and (0, 6). This distance is 1 units. (0, 6) (0, 3) a 1 Length of major ais 1 (0, 3) a 6 Divide each side b. The foci are located at (0, 3) and (0, 3), so c 3. We can use the relationship between a, b, and c to determine the value of b. (0, 6) c a b Equation relating a, b, and c 9 36 b c 3 and a 6 b 7 Solve for b. Since the major ais is vertical, substitute 36 for a and 7 for b in the form a 1. An equation of the ellipse is 3 b Chapter 8 Conic Sections

10 Eample Write an Equation Given the Lengths of the Aes MUSEUMS In an ellipse, sound or light coming from one focus is reflected to the other focus. In a whispering galler, a person can hear another person whisper from across the room if the two people are standing at the foci. The whispering galler at the Museum of Science and Industr in Chicago has an elliptical cross section that is 13 feet 6 inches b 7 feet inches. a. Write an equation to model this ellipse. Assume that the center is at the origin and the major ais is horizontal. The length of the major ais is or 1 feet. 3 a 1 Length of major ais Museums The whispering galler at Chicago s Museum of Science and Industr has a parabolic dish at each focus to help collect sound. Source: a 7 1 Divide each side b. 3 The length of the minor ais is 13 1 or 7 feet. b 7 Length of minor ais 7 b 7 Divide each side b. Substitute a 7 1 and b 7 into the form 3 a b 1. An equation of the ellipse is b. How far apart are the points at which two people should stand to hear each other whisper? People should stand at the two foci of the ellipse. The distance between the foci is c units. c a b c a b c a b c Substitute c 5.37 Equation relating a, b, and c Take the square root of each side. Multipl each side b. a = and b =. 3 Use a calculator. The points where two people should stand to hear each other whisper are about 5.37 feet or 5 feet inches apart. GRAPH ELLIPSES As with circles, ou can use completing the square, smmetr, and transformations to help graph ellipses. An ellipse with its center at the origin is represented b an equation of the form a b 1 or a 1. b The ellipse could be translated h units to the right and k units up. This would move the center to the point (h, k). Such a move would be equivalent to replacing with h and replacing with k. Equations of Ellipses with Centers at (h, k) Standard Form of Equation ( a h) ( b k) 1 ( a k) ( b h) 1 Direction of Major Ais horizontal vertical Foci (h c, k) (h, k c) Lesson 8- Ellipses 35

11 Eample 3 Graph an Equation in Standard Form Find the coordinates of the center and foci and the lengths of the major and minor aes of the ellipse with equation 1. Then graph the ellipse. 1 6 The center of this ellipse is at (0, 0). Since a 16, a. Since b, b. The length of the major ais is () or 8 units, and the length of the minor ais is () or units. Since the term has the greater denominator, the major ais is horizontal. c a b Equation relating a, b, and c c or 1 a, b c 1 or 3 Take the square root of each side. The foci are at 3, 0 and 3, 0. Stud Tip Graphing Calculator You can graph an ellipse on a graphing calculator b first solving for. Then graph the two equations that result on the same screen. You can use a calculator to find some approimate nonnegative values for and that satisf the equation. Since the ellipse is centered at the origin, it is smmetric about the -ais. Therefore, the points at (, 0), ( 3, 1.3), (, 1.7), and ( 1, 1.9) lie on the graph. The ellipse is also smmetric about the -ais, so the points at ( 3, 1.3), (, 1.7), ( 1, 1.9), (0, ), (1, 1.9), (, 1.7), and (3, 1.3) lie on the graph. Graph the intercepts, (, 0), (, 0), (0, ), and (0, ), and draw the ellipse that passes through them and the other points Chapter 8 Conic Sections If ou are given an equation of an ellipse that is not in standard form, write it in standard form first. This will make graphing the ellipse easier. Eample Graph an Equation Not in Standard Form Find the coordinates of the center and foci and the lengths of the major and minor aes of the ellipse with equation 0. Then graph the ellipse. Complete the square for each variable to write this equation in standard form. 0 riginal equation ( ) ( 6 ) ( ) Complete the squares. ( ) ( 6 9) (9), 6 9 ( ) ( 3) 16 Write the trinomials as perfect squares. ( ) ( 3) 1 Divide each side b

12 The graph of this ellipse is the graph from Eample 3 translated units to the left and up 3 units. The center is at (, 3) and the foci are at 3, 0 and 3, 0. The length of the major ais is still 8 units, and the length of the minor ais is still units. ( ) 16 ( 3) 1 You can use a circle to locate the foci on the graph of a given ellipse. Locating Foci You can locate the foci of an ellipse b using the following method. Step 1 Graph an ellipse so that its center is at the origin. Let the endpoints of the major 8 ais be at ( 9, 0) and (9, 0), and let the 6 endpoints of the minor ais be at (0, 5) and (0, 5). Step Step 3 Step Use a compass to draw a circle with center at (0, 0) and radius 9 units. Draw the line with equation 5 and mark the points at which the line intersects the circle Draw perpendicular lines from the points of intersection to the -ais. The foci of the ellipse are located at the points where the perpendicular lines intersect the -ais. Make a Conjecture Draw another ellipse and locate its foci. Wh does this method work? Concept Check 1. Identif the aes of smmetr of the ellipse at the right.. Eplain wh a circle is a special case of an ellipse. 3. PEN ENDED Write an equation for an ellipse with its center at (, 5) and a horizontal major ais. ( 1, 5) ( 1, 1) Guided Practice GUIDED PRACTICE KEY. Write an equation for the ellipse shown at the right. Write an equation for the ellipse that satisfies each set of conditions. 5. endpoints of major ais at (, ) and (, 10), endpoints of minor ais at (0, ) and (, ) ( 6, 0) (, 0) (, 0) (6, 0) 6. endpoints of major ais at (0, 10) and (0, 10), foci at (0, 8) and (0, 8) Lesson 8- Ellipses 37

13 Find the coordinates of the center and foci and the lengths of the major and minor aes for the ellipse with the given equation. Then graph the ellipse ( 1) ( ) Application 11. ASTRNMY At its closest point, Mercur is 9.0 million miles from the center of the Sun. At its farthest point, Mercur is 3.8 million miles from the center of the Sun. Write an equation for the orbit of Mercur, assuming that the center of the orbit is the origin and the Sun lies on the -ais. Practice and Appl Homework Help For Eercises See Eamples 1 1, , Etra Practice See page 86. Write an equation for each ellipse (0, 8) 6 (0, 5) (0, 5) (0, 8) (, 0) ( 3, 0) (3, 0) (, 0) (5 55, ) (5 55, ) (13, ) ( 3, ) (, ) (, ) (, 3) (, 3) Write an equation for the ellipse that satisfies each set of conditions. 16. endpoints of major ais at ( 11, 5) and (7, 5), endpoints of minor ais at (, 9) and (, 1) 17. endpoints of major ais at (, 1) and (, ), endpoints of minor ais at (, ) and (0, ) 18. major ais 0 units long and parallel to -ais, minor ais 6 units long, center at (, ) 19. major ais 16 units long and parallel to -ais, minor ais 9 units long, center at (5, ) 0. endpoints of major ais at (10, ) and ( 8, ), foci at (6, ) and (, ) 1. endpoints of minor ais at (0, 5) and (0, 5), foci at (1, 0) and ( 1, 0). INTERIR DESIGN The rounded top of the window is the top half of an ellipse. Write an equation for the ellipse if the origin is at the midpoint of the bottom edge of the window. 1 in. 36 in. 38 Chapter 8 Conic Sections

14 3. ASTRNMY At its closest point, Mars is 18.5 million miles from the Sun. At its farthest point, Mars is million miles from the Sun. Write an equation for the orbit of Mars. Assume that the center of the orbit is the origin, the Sun lies on the -ais, and the radius of the Sun is 00,000 miles.. WHITE HUSE There is an open area south of the White House known as the Ellipse. Write an equation to model the Ellipse. Assume that the origin is at the center of the Ellipse. 5. Write the equation 10 0 in standard form. The Ellipse 1057 ft 880 ft White House The Ellipse, also known as President s Park South, has an area of about 16 acres. Source: 6. What is the standard form of the equation ? Find the coordinates of the center and foci and the lengths of the major and minor aes for the ellipse with the given equation. Then graph the ellipse ( 8) ( ) ( 11) ( 5) CRITICAL THINKING Find an equation for the ellipse with foci at 3, 0 and 3, 0 that passes through (0, 3). 0. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. Wh are ellipses important in the stud of the solar sstem? Include the following in our answer: wh an equation that is an accurate model of the path of a planet might be useful, and the distance from the center of Earth s orbit to the center of the Sun given that the Sun is at a focus of the orbit of Earth. Use the information in the figure at the right. 91. million mi 9.5 million mi Nearest point Sun 00,000 mi Farthest point Standardized Test Practice 1. In the figure, A, B, and C are collinear. What is the measure of DBE? A 0 B 65 C 80 D E A B 55 C D Lesson 8- Ellipses 39

15 . 5 1 A 7 B 13 C 17 D 169 Etending the Lesson c 3. ASTRNMY In an ellipse, the ratio a is called the eccentricit and is denoted b the letter e. Eccentricit measures the elongation of an ellipse. As shown in the graph at the right, the closer e is to 0, the more an ellipse looks like a circle. Pluto has the most eccentric orbit in our solar sstem with e 0.5. Find an equation to model the orbit of Pluto, given that the length of the major ais is about 7.3 billion miles. Assume that the major ais is horizontal and that the center of the orbit is the origin. (e 0.7) (e 0) Maintain Your Skills Mied Review Write an equation for the circle that satisfies each set of conditions. (Lesson 8-3). center (3, ), radius 5 units 5. endpoints of a diameter at (5, 9) and (3, 11) 6. center ( 1, 0), passes through (, 6) 7. center (, 1), tangent to -ais 8. Write an equation of a parabola with verte (3, 1) and focus 3, 1 1. Then draw the graph. (Lesson 8-) MARRIAGE For Eercises 9 51, use the table at the right that shows the number of married Americans over the last few decades. (Lesson -5) 9. Draw a scatter plot in which is the number of ears since Find a prediction equation. 51. Predict the number of married Americans in 010. Number of Married Americans People (in millions) Year ? Source: U.S. Census Bureau nline Research Data Update For the latest statistics on marriage and other characteristics of the population, visit to learn more. Getting Read for the Net Lesson PREREQUISITE SKILL Graph the line with the given equation. (To review graphing lines, see Lessons -1, -, and -3.) ( 1) 57. ( 1) 0 Chapter 8 Conic Sections

16 Hperbolas Write equations of hperbolas. Graph hperbolas. Vocabular hperbola foci center verte asmptote transverse ais conjugate ais are hperbolas different from parabolas? A hperbola is a conic section with the propert that ras directed toward one focus are reflected toward the other focus. Notice that, unlike the other conic sections, a hperbola has two branches. EQUATINS F HYPERBLAS A hperbola is the set of all points in a plane such that the absolute value of the difference of the distances from two fied points, called the foci, is constant. The hperbola at the right has foci at (0, 3) and (0, 3). The distances from either of the -intercepts to the foci are 1 unit and 5 units, so the difference of the distances from an point with coordinates (, ) on the hperbola to the foci is or units, depending on the order in which ou subtract. You can use the Distance Formula and the definition of a hperbola to find an equation of this hperbola. (0, 3) (0, 3) (, ) The distance between the distance between (, ) and (0, 3) (, ) and (0, 3) = ±. ( 3) ( 3) ( 3) ( 3) ( 3) 16 8 ( 3) ( 3) ( 3) ( 3) 3 ( 3) 9 16 [ ( 3) ] Isolate the radicals. Square each side. Simplif. Divide each side b. Square each side. Distributive Propert 5 0 Simplif. 1 5 Divide each side b 0. An equation of this hperbola is 1. 5 Lesson 8-5 Hperbolas 1

17 The diagram below shows the parts of a hperbola. asmptote asmptote The point on each branch nearest the center is a verte. transverse center b ais c verte verte F 1 F a conjugate ais As a hperbola recedes from its center, the branches approach lines called asmptotes. A hperbola has some similarities to an ellipse. The distance from the center to a verte is a units. The distance from the center to a focus is c units. There are two aes of smmetr. The transverse ais is a segment of length a whose endpoints are the vertices of the hperbola. The conjugate ais is a segment of length b units that is perpendicular to the transverse ais at the center. The values of a, b, and c are related differentl for a hperbola than for an ellipse. For a hperbola, c a b. The table below summarizes man of the properties of hperbolas with centers at the origin. Stud Tip Reading Math In the standard form of a hperbola, the squared terms are subtracted ( ). For an ellipse, the are added ( ). Standard Form of Equation Equations of Hperbolas with Centers at the rigin a b 1 a b 1 Direction of Transverse Ais horizontal vertical Foci (c, 0), ( c, 0) (0, c), (0, c) Vertices (a, 0), ( a, 0) (0, a), (0, a) Length of Transverse Ais a units a units Length of Conjugate Ais b units b units Equations of Asmptotes b a b a Eample 1 Write an Equation for a Graph Write an equation for the hperbola shown at the right. The center is the midpoint of the segment connecting the vertices, or (0, 0). The value of a is the distance from the center to a verte, or 3 units. The value of c is the distance from the center to a focus, or units. (0, ) (0, 3) (0, 3) c a b Equation relating a, b, and c for a hperbola (0, ) 3 b c, a b Evaluate the squares. 7 b Solve for b. Since the transverse ais is vertical, the equation is of the form a b 1. Substitute the values for a and b. An equation of the hperbola is Chapter 8 Conic Sections

18 Navigation LRAN stands for Long Range Navigation. The LRAN sstem is generall accurate to within 0.5 nautical mile. Source: U.S. Coast Guard Eample Write an Equation Given the Foci and Transverse Ais NAVIGATIN The LRAN navigational sstem is based on hperbolas. Two stations send out signals at the same time. A ship notes the difference in the times at which it receives the signals. The ship is on a hperbola with the stations at the foci. Suppose a ship determines that the difference of its distances from two stations is 50 nautical miles. The stations are 100 nautical miles apart. Write an equation for a hperbola on which the ship lies if the stations are at ( 50, 0) and (50, 0). First, draw a figure. B studing either of the -intercepts, ou can see that the difference of the distances from an point on the hperbola to the stations at the foci is the same as the length of the transverse ais, or a. Therefore, a 50, or a 5. According to the coordinates of the foci, c 50. Use the values of a and c to determine the value of b for this hperbola. c a b 50 5 b c 50, a b Evaluate the squares b Solve for b. Equation relating a, b, and c for a hperbola Since the transverse ais is horizontal, the equation is of the form a 1. Substitute the values for a and b. An equation of the hperbola is ( 50, 0) (50, 0) b GRAPH HYPERBLAS So far, ou have studied hperbolas that are centered at the origin. A hperbola ma be translated so that its center is at (h, k). This corresponds to replacing b h and b k in both the equation of the hperbola and the equations of the asmptotes. Equations of Hperbolas with Centers at (h, k) Standard Form of Equation ( a h) ( b k) 1 ( a k) ( b h) 1 Direction of Transverse Ais horizontal vertical Equations of Asmptotes k b a ( h) k b a ( h) It is easier to graph a hperbola if the asmptotes are drawn first. To graph the asmptotes, use the values of a and b to draw a rectangle with dimensions a and b. The diagonals of the rectangle should intersect at the center of the hperbola. The asmptotes will contain the diagonals of the rectangle. Eample 3 Graph an Equation in Standard Form Find the coordinates of the vertices and foci and the equations of the asmptotes for the hperbola with equation 1. Then graph the hperbola. 9 The center of this hperbola is at the origin. According to the equation, a 9 and b, so a 3 and b. The coordinates of the vertices are (3, 0) and ( 3, 0). (continued on the net page) Lesson 8-5 Hperbolas 3

19 c a b Equation relating a, b, and c for a hperbola c 3 a 3, b c 13 c 13 Simplif. Take the square root of each side. The foci are at 13, 0 and 13, 0. The equations of the asmptotes are b a or 3. Stud Tip Graphing Calculator You can graph a hperbola on a graphing calculator. Similar to an ellipse, first solve the equation for. Then graph the two equations that result on the same screen. You can use a calculator to find some approimate nonnegative values for and that satisf the equation. Since the hperbola is centered at the origin, it is smmetric about the -ais. Therefore, the points at ( 8,.9), ( 7,.), ( 6, 3.5), ( 5,.7), (, 1.8), and ( 3, 0) lie on the graph. The hperbola is also smmetric about the -ais, so the points at ( 8,.9), ( 7,.), ( 6, 3.5), ( 5,.7), (, 1.8), (, 1.8), (5,.7), (6, 3.5), (7,.), and (8,.9) also lie on the graph. Draw a 6-unit b -unit rectangle. The asmptotes contain the diagonals of the rectangle. Graph the vertices, which, in this case, are the -intercepts. Use the asmptotes as a guide to draw the hperbola that passes through the vertices and the other points. The graph does not intersect the asmptotes When graphing a hperbola given an equation that is not in standard form, begin b rewriting the equation in standard form. Eample Graph an Equation Not in Standard Form Find the coordinates of the vertices and foci and the equations of the asmptotes for the hperbola with equation Then graph the hperbola. Complete the square for each variable to write this equation in standard form riginal equation ( 8 ) 9( ) 19 ( ) 9( ) Complete the squares. ( 8 16) 9( 1) 19 (16) 9(1) ( ) 9( 1) Write the trinomials as 36 perfect squares. ( ) ( 1) 1 Divide each side b The graph of this hperbola is the ( ) ( 1) graph from Eample 3 translated units 9 to the right and down 1 unit. The vertices are at (7, 1) and (1, 1), and the foci are at 13, 1 and 13, 1. The equations of the asmptotes are 1 ( ). 3 1 Chapter 8 Conic Sections

20 Concept Check Guided Practice GUIDED PRACTICE KEY 1. Determine whether the statement is sometimes, alwas, or never true. The graph of a hperbola is smmetric about the -ais.. Describe how the graph of k 1 changes as k increases. 3. PEN ENDED Find a countereample to the following statement. If the equation of a hperbola is a 1, then a b.. Write an equation for the hperbola shown at the right. 5. A hperbola has foci at (, 0) and (, 0). The value of a is 1. Write an equation for the hperbola. b (0, 5) (0, ) (0, ) (0, 5) Application Find the coordinates of the vertices and foci and the equations of the asmptotes for the hperbola with the given equation. Then graph the hperbola ( 6) 0 ( 1) ASTRNMY Comets that pass b Earth onl once ma follow hperbolic paths. Suppose a comet s path is modeled b a branch of the hperbola with equation 1. Find the coordinates of the vertices and foci and the 5 00 equations of the asmptotes for the hperbola. Then graph the hperbola. Practice and Appl Homework Help For Eercises See Eamples 11 0, 35 1, 1 3, 3, Etra Practice See page 86. Write an equation for each hperbola (, 0) (, 0) (, 0) (, 0) (, 3) (0, ) (0, 3) (3, 5) (0, 9) (0, 8) Lesson 8-5 Hperbolas 5

21 Write an equation for the hperbola that satisfies each set of conditions. 15. vertices ( 5, 0) and (5, 0), conjugate ais of length 1 units 16. vertices (0, ) and (0, ), conjugate ais of length 1 units 17. vertices (9, 3) and ( 5, 3), foci 53, vertices (, 1) and (, 9), foci, Find an equation for a hperbola centered at the origin with a horizontal transverse ais of length 8 units and a conjugate ais of length 6 units. 0. What is an equation for the hperbola centered at the origin with a vertical transverse ais of length 1 units and a conjugate ais of length units? Find the coordinates of the vertices and foci and the equations of the asmptotes for the hperbola with the given equation. Then graph the hperbola ( ) ( ) ( 3) ( ) ( 1) ( 3) 1 3. ( 6) ( 3) More About... Forester Foresters work for private companies or governments to protect and manage forest land. The also supervise the planting of trees and use controlled burning to clear weeds, brush, and logging debris. nline Research For information about a career as a forester, visit: careers FRESTRY For Eercises 35 and 36, use the following information. A forester at an outpost and another forester at the primar station both heard an eplosion. The outpost and the primar station are 6 kilometers apart. 35. If one forester heard the eplosion 6 seconds before the other, write an equation that describes all the possible locations of the eplosion. Place the two forester stations on the -ais with the midpoint between the stations at the origin. The transverse ais is horizontal. (Hint: The speed of sound is about 0.35 kilometer per second.) 36. Draw a sketch of the possible locations of the eplosion. Include the ranger stations in the drawing. 37. STRUCTURAL DESIGN An architect s design for a building includes some large pillars with cross sections in the shape of hperbolas. The curves can be modeled b the equation 1, where the units are in meters. If the pillars are meters tall, find the width of the top of each pillar and the width of each pillar at the narrowest point in the middle. Round to the nearest centimeter. 38. CRITICAL THINKING A hperbola with a horizontal transverse ais contains the point at (, 3). The equations of the asmptotes are 1 and 5. Write the equation for the hperbola. m 6 Chapter 8 Conic Sections

22 39. PHTGRAPHY A curved mirror is placed in a store for a wide-angle view of the room. The right-hand branch of 1 models 1 3 the curvature of the mirror. A small securit camera is placed so that all of the -foot diameter of the mirror is visible. If the back of the room lies on 18, what width of the back of the room is visible to the camera? (Hint: Find the equations of the lines through the focus and each edge of the mirror.) ft Standardized Test Practice 0. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How are hperbolas different from parabolas? Include the following in our answer: differences in the graphs of hperbolas and parabolas, and differences in the reflective properties of hperbolas and parabolas. 1. A leg of an isosceles right triangle has a length of 5 units. What is the length of the hpotenuse? A 5 units B 5 units C 5 units D 10 units. In the figure, what is the sum of the slopes of A B and A C? A 1 B 0 C 1 D 8 B(, 0) A C(, 0) Etending the Lesson A hperbola with asmptotes that are not perpendicular is called a nonrectangular hperbola. Most of the hperbolas ou have studied so far are nonrectangular. A rectangular hperbola has perpendicular asmptotes. For eample, the graph of 1 is a rectangular hperbola. The graphs of equations of the form c, where c is a constant, are rectangular hperbolas with the coordinate aes as their asmptotes. For Eercises 3 and, consider the equation. 3. Plot some points and use them to graph the equation. Be sure to consider negative values for the variables.. Find the coordinates of the vertices of the graph of the equation. 5. Graph. 6. Describe the transformations that can be applied to the graph of to obtain the graph of. 1 Maintain Your Skills Mied Review Write an equation for the ellipse that satisfies each set of conditions. (Lesson 8-) 7. endpoints of major ais at (1, ) and (9, ), endpoints of minor ais at (5, 1) and (5, 3) 8. major ais 8 units long and parallel to -ais, minor ais 6 units long, center at ( 3, 1) 9. foci at (5, ) and ( 3, ), major ais 10 units long Lesson 8-5 Hperbolas 7

23 50. Find the center and radius of the circle with equation Then graph the circle. (Lesson 8-3) Solve each equation b factoring. (Lesson 6-) q 11q 1 Perform the indicated operations, if possible. (Lesson -5) [1 3] PAGERS Refer to the graph at the right. What was the average rate of change of the number of pager subscribers from 1996 to 1999? (Lesson -3) 56. Solve 1 9. (Lesson 1-) 57. Simplif (Lesson 1-) USA TDAY Snapshots Staing in touch A new generation of pagers that can send and receive , news and other information from the Internet, is spurring industr growth. U.S. paging subscribers in millions: Source: Strategis Group for Personal Communications Association B Anne R. Care and Quin Tian, USA TDAY Getting Read for the Net Lesson PREREQUISITE SKILL Each equation is of the form A B C D E F 0. Identif the values of A, B, and C. (To review coefficients, see Lesson 5-1.) P ractice Quiz 1. Write an equation of the ellipse with foci at (3, 8) and (3, 6) and endpoints of the major ais at (3, 8) and (3, 10). (Lesson 8-) Find the coordinates of the center and foci and the lengths of the major and minor aes of the ellipse with the given equation. Then graph the ellipse. (Lesson 8-). ( ) ( ) Write an equation for the hperbola that satisfies each set of conditions. (Lesson 8-5). vertices ( 3, 0) and (3, 0), conjugate ais of length 8 units 5. vertices (, ) and (6, ), foci 1, Lessons 8- and Chapter 8 Conic Sections

24 Conic Sections Write equations of conic sections in standard form. Identif conic sections from their equations. can ou use a flashlight to make conic sections? Recall that parabolas, circles, ellipses, and hperbolas are called conic sections because the are the cross sections formed when a double cone is sliced b a plane. You can use a flashlight and a flat surface to make patterns in the shapes of conic sections. parabola circle ellipse hperbola Stud Tip Reading Math In this lesson, the word ellipse means an ellipse that is not a circle. STANDARD FRM The equation of an conic section can be written in the form of the general quadratic equation A B C D E F 0, where A, B, and C are not all zero. If ou are given an equation in this general form, ou can complete the square to write the equation in one of the standard forms ou have learned. Conic Section Parabola Standard Form of Conic Sections Standard Form of Equation a( h) k or a( k) h Circle ( h) ( k) r Ellipse ( a h) ( b k) 1 or ( a k) ( b h) 1, a b Hperbola ( a h) ( b k) 1 or ( a k) ( b h) 1 Eample 1 Rewrite an Equation of a Conic Section Write the equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola. Then graph the equation. Write the equation in standard form riginal equation 6 7 Isolate terms Complete the square. ( 3) ( 3) ( 3) 1 Divide each side b The graph of the equation is an ellipse with its center at (3, 0). ( 3) Lesson 8-6 Conic Sections 9

25 IDENTIFY CNIC SECTINS Instead of writing the equation in standard form, ou can determine what tpe of conic section an equation of the form A B C D E F 0, where B 0, represents b looking at A and C. Conic Section Parabola Circle Identifing Conic Sections Relationship of A and C A 0 or C 0, but not both. A C Ellipse A and C have the same sign and A C. Hperbola A and C have opposite signs. Eample Analze an Equation of a Conic Section Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hperbola. a. 0 A and C 1. Since A and C have opposite signs, the graph is a hperbola. b A and C. Since A C, the graph is a circle. c C 1. Since there is no term, A 0. The graph is a parabola. Concept Check Guided Practice GUIDED PRACTICE KEY Application 1. PEN ENDED Write an equation of the form A B C D E F 0, where A, that represents a circle.. Write the general quadratic equation for which A, B 0, C 0, D, E 7, and F Eplain wh the graph of 5 0 is a single point. Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola. Then graph the equation Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hperbola AVIATIN For Eercises 10 and 11, use the following information. When an airplane flies faster than the speed of sound, it produces a shock wave in the shape of a cone. Suppose the shock wave intersects the ground in a curve that can be modeled b Identif the shape of the curve. 11. Graph the equation. 50 Chapter 8 Conic Sections

26 Practice and Appl Homework Help For Eercises See Eamples Etra Practice See page 86. Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola. Then graph the equation ( 1) 9( ) ( ) 18. ( ) 9( ) ( ) 7. (8 3) ASTRNMY The orbits of comets follow paths in the shapes of conic sections. For eample, Halle s Comet follows an elliptical orbit with the Sun located at one focus. What tpe(s) of orbit(s) pass b the Sun onl once? WATER For Eercises 31 and 3, use the following information. If two stones are thrown into a lake at different points, the points of intersection of the resulting ripples will follow a conic section. Suppose the conic section has the equation Identif the shape of the curve. 3. Graph the equation. Parabolic orbit rbiting object Elliptical orbit Central bod Hperbolic orbit Circular orbit Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hperbola ( ) 39. Identif the shape of the graph of the equation 3 0. Astronom Halle s Comet orbits the Sun about once ever 76 ears. It will return net in 061. Source: What tpe of conic section is represented b the equation 6 8? For Eercises 1 3, match each equation below with the situation that it could represent. a b c SPRTS the flight of a baseball. PHTGRAPHY the oval opening in a picture frame 3. GEGRAPHY the set of all points that are 0 miles from a landmark Lesson 8-6 Conic Sections 51

27 CRITICAL THINKING For Eercises and 5, use the following information. The graph of an equation of the form a 0 is a special case of a hperbola.. Identif the graph of such an equation. 5. Eplain how to obtain such a set of points b slicing a double cone with a plane. 6. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How can ou use a flashlight to make conic sections? Include the following in our answer: an eplanation of how ou could point the flashlight at a ceiling or wall to make a circle, and an eplanation of how ou could point the flashlight to make a branch of a hperbola. b Standardized Test Practice 7. Which conic section is not smmetric about the -ais? A 3 0 B 1 0 C D What is the equation of the graph at the right? A 1 B 1 C 1 D 1 Etending the Lesson 9. Refer to Eercise 3 on page 0. Eccentricit can be studied for conic sections c other than ellipses. The epression for the eccentricit of a hperbola is, just a as for an ellipse. The eccentricit of a parabola is 1. Find inequalities for the eccentricities of noncircular ellipses and hperbolas, respectivel. Maintain Your Skills Mied Review Write an equation of the hperbola that satisfies each set of conditions. (Lesson 8-5) 50. vertices (5, 10) and (5, ), conjugate ais of length 8 units 51. vertices (6, 6) and (0, 6), foci 3 13, 6 5. Find the coordinates of the center and foci and the lengths of the major and minor aes of the ellipse with equation Then graph the ellipse. (Lesson 8-) Getting Read for the Net Lesson Simplif. Assume that no variable equals 0. (Lesson 5-1) 53. ( 3 ) 5. (m 5 n 3 ) m n HEALTH The prediction equation relates a person s maimum heart rate for eercise and age. Use the equation to find the maimum heart rate for an 18-ear-old. (Lesson -5) PREREQUISITE SKILL Solve each sstem of equations. (To review solving sstems of linear equations, see Lesson 3-.) Chapter 8 Conic Sections

28 A Follow-Up of Lesson 8-6 Conic Sections Recall that a parabola is the set of all points that are equidistant from the focus and the directri. You can draw a parabola based on this definition b using special conic graph paper. This graph paper contains a series of concentric circles equall spaced from each other and a series of parallel lines tangent to the circles. Number the circles consecutivel beginning with the smallest circle. Number the lines with consecutive integers as shown in the sample at the right. Be sure that line 1 is tangent to circle Activit 1 Mark the point at the intersection of circle 1 and line 1. Mark both points that are on line and circle. Continue this process, marking both points on line 3 and circle 3, and so on. Then connect the points with a smooth curve. Look at the diagram at the right. What shape is the graph? Note that ever point on the graph is equidistant from the center of the small circle and the line labeled 0. The center of the small circle is the focus of the parabola, and line 0 is the directri Algebra Activit Conic Sections 53

29 Algebra Activit Activit An ellipse is the set of points such that the sum of the distances from two fied points is constant. The two fied points are called the foci. Use graph paper like that shown. It contains two small circles and a series of concentric circles from each. The concentric circles are tangent to each other as shown. Choose the constant 13. Mark the points at the intersections of circle 9 and circle, because Continue this process until ou have marked the intersection of all circles whose sum is 13. Connect the points to form a smooth curve. The curve is an ellipse whose foci are the centers of the two small circles on the graph paper Activit 3 A hperbola is the set of points such that the difference of the distances from two fied points is constant. The two fied points are called the foci. Use the same tpe of graph paper that ou used for the ellipse in Activit. Choose the constant 7. Mark the points at the intersections of circle 9 and circle, because 9 7. Continue this process until ou have marked the intersections of all circles whose difference in radius is 7. Connect the points to form a hperbola Model and Analze 1. Use the tpe of graph paper ou used in Activit 1. Mark the intersection of line 0 and circle. Then mark the two points on line 1 and circle 3, the two points on line and circle, and so on. Draw the new parabola. Continue this process and make as man parabolas as ou can on one sheet of the graph paper. The focus is alwas the center of the small circle. Wh are the resulting graphs parabolas?. In Activit, ou drew an ellipse such that the sum of the distances from two fied points was 13. Choose 10, 11, 1, 1, and so on for that sum, and draw as man ellipses as ou can on one piece of the graph paper. a. Wh can ou not start with 9 as the sum? b. What happens as the sum increases? decreases? 3. In Activit 3, ou drew a hperbola such that the difference of the distances from two fied points was 7. Choose other numbers and draw as man hperbolas as ou can on one piece of graph paper. What happens as the difference increases? decreases? 5 Chapter 8 Conic Sections

30 Solving Quadratic Sstems Solve sstems of quadratic equations algebraicall and graphicall. Solve sstems of quadratic inequalities graphicall. do sstems of equations appl to video games? Computer software often uses a coordinate sstem to keep track of the locations of objects on the screen. Suppose an enem space station is located at the center of the screen, which is the origin in a coordinate sstem. The space station is surrounded b a circular force field of radius 50 units. If the spaceship ou control is fling toward the center along the line with equation 3, the point where the ship hits the force field is a solution of a sstem of equations. SYSTEMS F QUADRATIC EQUATINS If the graphs of a sstem of equations are a conic section and a line, the sstem ma have zero, one, or two solutions. Some of the possible situations are shown below. no solutions one solution two solutions You have solved sstems of linear equations graphicall and algebraicall. You can use similar methods to solve sstems involving quadratic equations. Eample 1 Linear-Quadratic Sstem Solve the sstem of equations. 9 3 You can use a graphing calculator to help visualize the relationships of the graphs of the equations and predict the number of solutions. Solve each equation for to obtain 9 and 1 3. Enter the functions 9 9,, and 1 3 on the Y screen. The graph indicates that the hperbola and line intersect in two points. So the sstem has [ 10, 10] scl: 1 b [ 10, 10] scl: 1 two solutions. (continued on the net page) Lesson 8-7 Solving Quadratic Sstems 55

31 Use substitution to solve the sstem. First rewrite 3 as 3. 9 First equation in the sstem ( 3) 9 Substitute 3 for. 1 0 Simplif. 0 Divide each side b 1. ( ) 0 Factor. 0 or 0 Zero Product Propert Now solve for. Solve for. 3 3 Equation for in terms of (0) 3 () 3 Substitute the values. 3 5 Simplif. The solutions of the sstem are ( 3, 0) and (5, ). Based on the graph, these solutions are reasonable. If the graphs of a sstem of equations are two conic sections, the sstem ma have zero, one, two, three, or four solutions. Some of the possible situations are shown below. no solutions one solution two solutions three solutions four solutions Stud Tip Graphing Calculators If ou use ZSquare on the ZM menu, the graph of the first equation will look like a circle. Eample Quadratic-Quadratic Sstem Solve the sstem of equations A graphing calculator indicates that the circle and ellipse intersect in four points. So, this sstem has four solutions. Use the elimination method to solve the sstem Rewrite the first original equation. ( ) 5 Second original equation 3 1 Add. Divide each side b 3. [ 10, 10] scl: 1 b [ 10, 10] scl: 1 Take the square root of each side. 56 Chapter 8 Conic Sections

32 Substitute and for in either of the original equations and solve for. 5 5 Second original equation () 5 ( ) 5 Substitute for. 9 9 Subtract 16 from each side. 3 3 Take the square root of each side. The solutions are (3, ), ( 3, ), ( 3, ), and (3, ). A graphing calculator can be used to approimate the solutions of a sstem of equations. Quadratic Sstems The calculator screen shows the graphs of two circles. Think and Discuss 1. Write the sstem of equations represented b the graph.. Enter the equations into a TI-83 Plus and use the intersect feature on the CALC menu to solve the sstem. Round to the nearest hundredth. 3. Solve the sstem algebraicall.. Can ou alwas find the eact solution of a [ 10, 10] scl: 1 b [ 10, 10] scl: 1 sstem using a graphing calculator? Eplain. Use a graphing calculator to solve each sstem of equations. Round to the nearest hundredth Stud Tip Graphing Quadratic Inequalities If ou are unsure about which region to shade, ou can test one or more points, as ou did with linear inequalities. SYSTEMS F QUADRATIC INEQUALITIES You have learned how to solve sstems of linear inequalities b graphing. Sstems of quadratic inequalities are also solved b graphing. The graph of an inequalit involving a parabola, circle, or ellipse is either the interior or the eterior of the conic section. The graph of an inequalit involving a hperbola is either the region between the branches or the two regions inside the branches. As with linear inequalities, eamine the inequalit smbol to determine whether to include the boundar. Eample 3 Sstem of Quadratic Inequalities Solve the sstem of inequalities b graphing. 16 The graph of is the parabola and the region outside or below it. This region is shaded blue. The graph of 16 is the interior of the circle 16. This region is shaded ellow. The intersection of these regions, shaded green, represents the solution of the sstem of inequalities Lesson 8-7 Solving Quadratic Sstems 57

33 Concept Check Guided Practice GUIDED PRACTICE KEY 1. Graph each sstem of equations. Use the graph to solve the sstem. a. 3 0 b Sketch a parabola and an ellipse that intersect at eactl three points. 3. PEN ENDED Write a sstem of quadratic equations for which (, 6) is a solution. Find the eact solution(s) of each sstem of equations Solve each sstem of inequalities b graphing Application 10. EARTHQUAKES In a coordinate sstem where a unit represents one mile, the epicenter of an earthquake was determined to be 50 miles from a station at the origin. It was also 0 miles from a station at (0, 30) and 13 miles from a station at (35, 18). Where was the epicenter located? Practice and Appl Homework Help For Eercises See Eamples 11 16, 5, , 6 8, 30, Etra Practice See page Chapter 8 Conic Sections Find the eact solution(s) of each sstem of equations Where do the graphs of the equations 1 and 11 intersect? 6. What are the coordinates of the points that lie on the graphs of both 5 and 3 66? 7. RCKETS Two rockets are launched at the same time, but from different heights. The height in feet of one rocket after t seconds is given b 16t 150t 5. The height of the other rocket is given b 16t 160t. After how man seconds are the rockets at the same height?

34 8. ADVERTISING The corporate logo for an automobile manufacturer is shown at the right. Write a sstem of three equations to model this logo. 9. MIRRRS A hperbolic mirror is a mirror in the shape of one branch of a hperbola. Such a mirror reflects light ras directed at one focus toward the other focus. Suppose a hperbolic mirror is modeled b the upper branch of the hperbola with equation 1. A light source is located at ( 10, 0) Where should the light from the source hit the mirror so that the light will be reflected to (0, 5)? ASTRNMY For Eercises 30 and 31, use the following information. The orbit of Pluto can be modeled b the equation , where the units are astronomical units. Suppose a comet is following a path modeled b the equation Find the point(s) of intersection of the orbits of Pluto and the comet. Round to the nearest tenth. 31. Will the comet necessaril hit Pluto? Eplain. Astronom The astronomical unit (AU) is the mean distance between Earth and the Sun. ne AU is about 93 million miles or 150 million kilometers. Source: Solve each sstem of inequalities b graphing CRITICAL THINKING For Eercises 38, find all values of k for which the sstem of equations has the given number of solutions. If no values of k meet the condition, write none. k no solutions 39. one solution 0. two solutions 1. three solutions. four solutions Standardized Test Practice 3. WRITING IN MATH Answer the question that was posed at the beginning of the lesson. How do sstems of equations appl to video games? Include the following in our answer: a linear-quadratic sstem of equations that applies to this situation, an eplanation of how ou know that the spaceship is headed directl toward the center of the screen, and the coordinates of the point at which the spaceship will hit the force field, assuming that the spaceship moves from the bottom of the screen toward the center.. If is defined to be for all numbers, which of the following is the greatest? A 0 B 1 C D 3 5. How man three-digit numbers are divisible b 3? A 99 B 300 C 301 D 30 Lesson 8-7 Solving Quadratic Sstems 59

35 Graphing Calculator SYSTEMS F EQUATINS Write a sstem of equations that satisfies each condition. Use a graphing calculator to verif that ou are correct. 6. two parabolas that intersect in two points 7. a hperbola and a circle that intersect in three points 8. a circle and an ellipse that do not intersect 9. a circle and an ellipse that intersect in four points 50. a hperbola and an ellipse that intersect in two points 51. two circles that intersect in three points Maintain Your Skills Mied Review Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola. Then graph the equation. (Lesson 8-6) Find the coordinates of the vertices and foci and the equations of the asmptotes of the hperbola with the equation 6. Then graph the hperbola. (Lesson 8-5) Solve each equation b factoring. (Lesson 6-5) For Eercises 61 and 6, complete parts a c for each quadratic equation. (Lesson 6-5) a. Find the value of the discriminant. b. Describe the number and tpe of roots. c. Find the eact solutions b using the Quadratic Formula Simplif. (Lesson 5-9) 63. (3 i) (1 7i) 6. (8 i)( 3i) 65. 3i 1 i 66. CHEMISTRY The mass of a proton is about kilogram. The mass of an electron is about kilogram. About how man times as massive as an electron is a proton? (Lesson 5-1) Evaluate each determinant. (Lesson -3) Solve the sstem of equations. (Lesson 3-5) r s t 15 r t 1 s t 10 Write an equation in slope-intercept form for each graph. (Lesson -) (, ) (1, 1) (, ) (1, 3) 60 Chapter 8 Conic Sections

36 Vocabular and Concept Check asmptote (p. ) center of a circle (p. 6) center of a hperbola (p. ) center of an ellipse (p. 3) circle (p. 6) conic section (p. 19) conjugate ais (p. ) directri (p. 19) Distance Formula (p. 13) ellipse (p. 33) foci of a hperbola (p. 1) foci of an ellipse (p. 33) focus of a parabola (p. 19) hperbola (p. 1) latus rectum (p. 1) major ais (p. 3) Midpoint Formula (p. 1) minor ais (p. 3) parabola (p. 19) tangent (p. 7) transverse ais (p. ) verte of a hperbola (p. ) Tell whether each statement is true or false. If the statement is false, correct it to make it true. 1. An ellipse is the set of all points in a plane such that the sum of the distances from two given points in the plane, called the foci, is constant.. The major ais is the longer of the two aes of smmetr of an ellipse. 3. The formula used to find the distance between two points in a coordinate plane is d ( 1 ) (. 1 ). A parabola is the set of all points that are the same distance from a given point called the directri and a given line called the focus. 5. The radius is the distance from the center of a circle to an point on the circle. 6. The conjugate ais of a hperbola is a line segment parallel to the transverse ais. 7. A conic section is formed b slicing a double cone b a plane. 8. A hperbola is the set of all points in a plane such that the absolute value of the sum of the distances from an point on the hperbola to two given points is constant. 9. The midpoint formula is given b 1, The set of all points in a plane that are equidistant from a given point in a plane, called the center, forms a circle. 8-1 See pages Eamples Midpoint and Distance Formulas Concept Summar Midpoint Formula: M 1, 1 Distance Formula: d ( 1 ) ( 1 ) 1 Find the midpoint of a segment whose endpoints are at ( 5, 9) and (11, 1). 1, , 9 ( 1) Let ( 1, 1 ) ( 5, 9) and (, ) (11, 1). 6, 8 or (3, ) Simplif. ( 1, 1 ) d M (, ) Chapter 8 Stud Guide and Review 61

37 Chapter 8 Stud Guide and Review Find the distance between P(6, ) and Q( 3, 8). d ( 1 ) ( 1 ) Distance Formula 6) [8 ( ( )] Let ( 1, 1 ) (6, ) and (, ) ( 3, 8). Subtract. 5 or 15 units Simplif. Eercises Find the midpoint of the line segment with endpoints at the given coordinates. See Eample 1 on page (1, ), (, 6) 1. ( 8, 0), (, 3) , 7, 1, 5 Find the distance between each pair of points with the given coordinates. See Eamples and 3 on pages 13 and (, 10), (, 13) 15. (8, 5), ( 9, ) 16. (7, 3), (1, ) 8- See pages Eample Parabolas Concept Summar Standard Form Verte Ais of Smmetr Focus Directri Graph 1 7. First write the equation in the form a( h) k. 1 7 Parabolas a( h) k a( k) h (h, k) (h, k) h k 1 h, k 1a h, a k k 1 h 1 a a riginal equation 1 7 Isolate the terms with. ( 1 ) 7 Complete the square. ( 1 9) 7 9 Add and subtract 9, since 1 9. ( 7) ( 7) 1 ( 7) 19 Divide each side b. verte: ( 7, 19) focus: 19 7, 1 1 ais of smmetr: 7 or ( 7, 18) 1 directri: 19 or 0 1 direction of opening: upward since a Chapter 8 Conic Sections

38 Chapter 8 Stud Guide and Review Eercises Identif the coordinates of the verte and focus, the equations of the ais of smmetr and directri, and the direction of opening of the parabola with the given equation. Then find the length of the latus rectum and graph the parabola. See Eamples on pages ( 1) 1( 1) ( 3) Write an equation for a parabola with verte (0, 1) and focus (0, 1). Then graph the parabola. See Eample on pages and Circles See pages Concept Summar The equation of a circle with center (h, k) and radius r can be written in the form ( h) ( k) r. Eample Graph First write the equation in the form ( h) ( k) r riginal equation 8 16 Complete the squares , 1 ( ) ( 1) 1 Write the trinomials as squares. The center of the circle is at (, 1) and the radius is 1. Now draw the graph Eercises Write an equation for the circle that satisfies each set of conditions. See Eample 1 on page 6.. center (, 3), radius 5 units 3. center (, 0), radius 3 unit. endpoints of a diameter at (9, ) and ( 3, ) 5. center at ( 1, ), tangent to -ais Find the center and radius of the circle with the given equation. Then graph the circle. See Eamples and 5 on page ( 5) ( 11) Chapter 8 Stud Guide and Review 63

39 Chapter 8 Stud Guide and Review 8- See pages Ellipses Concept Summar Standard Form of Equation Direction of Major Ais ( a h) Ellipses ( b k) 1 ( a k) horizontal ( h) 1 b vertical Eample Graph First write the equation in standard form b completing the squares ( 8 ) 31 3( ) riginal equation Complete the squares ( 8 16) (16) 16 6, 8 16 ( 8) 3( ) 81 Write the trinomials as squares. ( 8) ( ) 1 Divide each side b The center of the ellipse is at (8, ). The length of the major ais is 18, and the length of the minor ais is Eercises 30. Write an equation for the ellipse with endpoints of the major ais at (, 1) and ( 6, 1) and endpoints of the minor ais at ( 1, 3) and ( 1, 1). See Eamples 1 and on pages 3 and 35. Find the coordinates of the center and foci and the lengths of the major and minor aes for the ellipse with the given equation. Then graph the ellipse. See Eamples 3 and on pages 36 and ( ) 16 ( 3) Hperbolas See pages Concept Summar 1 8. Standard Form Hperbolas ( a h) ( b k) 1 ( a k) ( b h) 1 Transverse Ais horizontal vertical Asmptotes k b a ( h) k a ( h) b 6 Chapter 8 Conic Sections

40 Chapter 8 Stud Guide and Review Eample Graph Complete the square for each variable to write this equation in standard form riginal equation 9( ) ( 8 ) 91 9( ) ( ) Complete the squares. 9( 1) ( 8 16) 91 9(1) (16) 1, ( 1) ( ) 36 Write the trinomials as squares. ( 1) ( ) 1 Divide each side b The center is at ( 1, ). The vertices are at ( 3, ) and (1, ) and the foci are at 1 13,. The equations of the asmptotes are 3 ( 1). ( 1) ( ) 1 9 Eercises 3. Write an equation for a hperbola that has vertices at (, 5) and (, 1) and a conjugate ais of length 6 units. See Eample 1 on page. Find the coordinates of the vertices and foci and the equations of the asmptotes for the hperbola with the given equation. Then graph the hperbola. See Eamples 3 and on pages 3 and ( ) ( 1) See pages 9 5. Conic Sections Concept Summar Conic sections can be identified directl from their equations of the form A B C D E F 0, assuming B 0. Conic Section Relationship of A and C Parabola A 0 or C 0, but not both. Circle A C Ellipse A and C have the same sign and A C. Hperbola A and C have opposite signs. Eample Without writing the equation in standard form, state whether the graph of is a parabola, circle, ellipse, or hperbola. In this equation, A and C 9. Since A and C are both positive and A C, the graph is an ellipse. Chapter 8 Stud Guide and Review 65

41 Etra Practice, see pages Mied Problem Solving, see page 869. Eercises Write each equation in standard form. State whether the graph of the equation is a parabola, circle, ellipse, or hperbola. Then graph the equation. See Eample 1 on page Without writing the equation in standard form, state whether the graph of each equation is a parabola, circle, ellipse, or hperbola. See Eample on page See pages Eample Solving Quadratic Sstems Concept Summar Sstems of quadratic equations can be solved using substitution and elimination. A sstem of quadratic equations can have zero, one, two, three, or four solutions. Solve the sstem of equations Use substitution to solve the sstem. First, rewrite 0 as First original equation ( ) 1( ) 1 0 Substitute for Simplif Divide each side b. ( 6)( 1) 0 Factor. 6 0 or 1 0 Zero Product Propert. 6 1 Solve for. Now solve for. Equation for in terms of ( 6) or 6 ( 1) or 1 Substitute the values. The solutions of the sstem are ( 6, 6) and ( 1, 1). Eercises Find the eact solution(s) of each sstem of equations. See Eamples 1 and on pages Solve each sstem of inequalities b graphing. See Eample 3 on page Chapter 8 Conic Sections

42 Vocabular and Concepts Choose the letter that best matches each description. 1. the set of all points in a plane that are the same distance from a given point, the focus, and a given line, the directri. the set of all points in a plane such that the absolute value of the difference of the distances from two fied points, the foci, is constant 3. the set of all points in a plane such that the sum of the distances from two fied points, the foci, is constant a. ellipse b. parabola c. hperbola Skills and Applications Find the midpoint of the line segment with endpoints at the given coordinates.. (7, 1), ( 5, 9) , 1, 8 5, 6. ( 13, 0), ( 1, 8) Find the distance between each pair of points with the given coordinates. 7. ( 6, 7), (3, ) 8. 1, 5, 3, (8, 1), (8, 9) State whether the graph of each equation is a parabola, circle, ellipse, or hperbola. Then graph the equation ( ) 7( 5) ( ) TUNNELS The opening of a tunnel is in the shape of a semielliptical arch. The arch is 60 feet wide and 0 feet high. Find the height of the arch 1 feet from the edge of the tunnel. 0 ft 60 ft Find the eact solution(s) of each sstem of equations Solve the sstem of inequalities b graphing STANDARDIZED TEST PRACTICE Which is not the equation of a parabola? A 9 B C 8 8 D 1 ( 1) 5 Chapter 8 Practice Test 67

43 Part 1 Record our answers on the answer sheet provided b our teacher or on a sheet of paper. 1. The product of a prime number and a composite number must be A C prime. even. composite. negative.. In 1990, the population of Claton was 5,00, and the population of Montrose was 7,500. B 000, the population of each cit had decreased b eactl 5%. How man more people lived in Claton than in Montrose in 000? A C Multiple Choice 335 B D If % of n is equal to 0% of p, then n is what percent of 10p? A 1 % B 10% C 100% D 1,000% B D 5. What is the midpoint of the line segment whose endpoints are at ( 5, 3) and ( 1, )? A 3, 1 B 3, 1 C, 7 D, 1 6. Point M(, 3) is the midpoint of line segment NP. If point N has coordinates ( 7, 1), then what are the coordinates of point P? A C ( 5, ) B (, 6) 9, D (3, 5) 7. Which equation s graph is a parabola? A B C D What is the center of the circle with equation 6 9 0? A C (, 6) B (, 3) (, 3) D (3, 3). Lero bought m magazines at d dollars per magazine and p paperback books at d 1 dollars per book. Which of the following represents the total amount Lero spent? 9. What is the distance between the points shown in the graph? A C d(m p) p B (m p)(3d 1) md pd 1 D pd(m ) Test-Taking Tip Questions 3, In problems with variables, ou can substitute values to tr to eliminate some of the answer choices. For eample, in Question 3, choose a value for n and compute the corresponding value n of p. Then find to answer the question. 1 0p A B C D 3 units 5 units 3 units 17 units 10. The median of seven test scores is 5, the mode is 6, the lowest score is 0, and the average is 53. If the scores are integers, what is the greatest possible test score? A 68 B 7 C 76 D 8 68 Chapter 8 Conic Sections

44 Aligned and verified b Part Short Response/Grid In Part 3 Quantitative Comparison Record our answers on the answer sheet provided b our teacher or on a sheet of paper. Compare the quantit in Column A and the quantit in Column B. Then determine whether: 11. What is the least positive integer p for which p 3 is not a prime number? 1. The ratio of cars to SUVs in a parking lot is to 5. After 6 cars leave the parking lot, the ratio of cars to SUVs becomes 1 to. How man SUVs are in the parking lot? A B C D the quantit in Column A is greater, the quantit in Column B is greater, the two quantities are equal, or the relationship cannot be determined from the information given. 13. Each dimension of a rectangular bo is an integer greater than 1. If the area of one side of the bo is 7 square units and the area of another side is 1 square units, what is the volume of the bo in cubic units? 1. Let the operation * be defined as a * b ab (a b). If * 10, then what is the value of? Column A 18. Tangent circles R, S, and T each have an area of 16 square units. Column B R S T 15. If the slope of line PQ in the figure is 1, perimeter of RST circumference of R what is the area of quadrilateral PQR? Q P(0, 3) 19. the ratio of girls to bos in Class A that contains more bos than girls the ratio of girls to bos in Class B that contains more girls than bos R(, 0) 0. Percent of Lakewood School Students Living in Each Town 16. In the figure, the slope of line is 5, and the slope of line k is 3. What is the distance 8 from point A to point B? A(, 1 ) Langton Haward 30% Delton 6% Cedargrove 0% B(, ) 17. If ( 3)( n) a b 15 for all values of, what is the value of a b? k the number of students who do not live in Langton the number of students who do not live in Delton 1. nk 10 n and k are positive integers. nk n k Chapter 8 Standardized Test Practice 69