Lesson 1-6 Lesson 1-6 BIG IDEA From the geometric defi nition of a hperbola, an equation for an hperbola smmetric to the - and -aes can be found. The edges of the silhouettes of each of the towers pictured at the right are parts of hperbolas. Structures with this shape are able to withstand higher winds and require less material to build than an other form. What Is a Hperbola? Like an ellipse, a hperbola is determined b two foci and a focal constant. However, instead of a constant sum of distances from the foci, a point on a hperbola must be at a constant difference of distances from the foci. The following Activit shows one wa to find points on a hperbola. Activit MATERIALS conic graph paper with 6 units between the centers of the circles Step 1 Cop the foci and points P 1 and P at the right. Find P 1, P, P 1, and P, then calculate P 1 - P 1 and P - P. Do both differences equal the same constant? Step Plot two more points P n such that P n = 8 and P n = 6, and then two more such that P n = 7 and P n = 5. Continue this process to fi nd four more points such that P n - P n is alwas. (continued on net page) Equations for Some Hperbolas 6 7 8 5 P 1 4 1 1 4 5 6 P 8 7 Vocabular hperbola foci, focal constant of a hperbola vertices of a hperbola standard position of a hperbola standard form of an equation for a hperbola Mental Math Suppose a function f contains the points (4, 17), (9, 1), and (1, 1). a. Find the rate of change from (4, 17) to (9, 1). b. Find the rate of change from (9, 1) to (1, 1). c. Could the graph of f be a line? Equations for Some Hperbolas 81
Chapter 1 Step Repeat Step, plotting ten points P n such that P n - P n =. Step 4 Draw a smooth curve through the points ou plotted in Step, and another through the points ou plotted in Step. These are two branches of a hperbola. The branches do not intersect. In general, if d is a positive number less than, the set of all points P such that P - P = d is a hperbola. The absolute value means that the hperbola has two branches, one from P - P = d, and the other from P - P = d. The absolute value function allows both branches to be described with one equation. Definition of Hperbola Let and be an two points and d be a constant with 0 < d <. Then the hperbola with foci and and focal constant d is the set of points P in a plane that satisf P - P = d. The vertices V 1 and V of the hperbola are the intersection points of F 1 and the hperbola. While it ma look like each branch of the hperbola is a parabola, this is not the case. Each branch of a hperbola has asmptotes. In the figure at the right, l 1 and l are asmptotes. The farther points on the hperbola are from a verte of the hperbola, the closer the are to an asmptote, without ever touching. In contrast, parabolas do not have asmptotes. The Standard Form of an Equation for a Hperbola A hperbola is in standard position if it is centered at the origin with its foci on an ais. An equation for a hperbola in standard position resembles the standard form of an equation for an ellipse. Equation for a Hperbola Theorem branch P The hperbola with foci (c, 0) and ( c, 0) and focal constant a has equation _ a - _ b = 1, where b = c - a. verte V 1 verte V P - P = d l 1 l asmptote asmptote branch 8 Quadratic Relations
Lesson 1-6 Proof The proof is almost identical to the proof of the Equation for an Ellipse Theorem in Lesson 1-4. Let P = (, ) be an point on the hperbola with foci = ( c, 0) and = (c, 0) and focal constant a. Then, b the defi nition of a hperbola, P - P = a. B the defi nition of absolute value, ou know that this equation is equivalent to P - P = ± a. Now substitute P = (, ), = ( c, 0), and = (c, 0) into the Pthagorean Distance Formula to get ( + c) + ( - 0) - ( - c) + ( - 0) = ± a. Do algebraic manipulations similar to those in Steps 1 9 of the proof in Lesson 1-4, and the same equation in Step 9 results. (a - c ) + a = a (a - c ) Then in Step 10, for hperbolas, c > a > 0, so c > a. Thus, c - a is positive and ou can let b = c - a. So b = a - c. This accounts for the minus sign in the equation. _ a - _ b = 1 The equation _ a - _ = 1 is the standard form of an equation for b a hperbola. = ( c, 0) ( a, 0) 0 (a, 0) = (c, 0) P = (, ) Eample 1 Find an equation for the hperbola with foci and, where = 10 and P - P = 8, on a rectangular coordinate sstem in standard position. P P 1 Solution Use the Equation for a Hperbola Theorem. You are given = 10, so c = 10, and c = 5. The focal constant is 8, so a = 8, and a = 4. Now, b = 5-4 = 9. Thus, an equation for this hperbola is 16-9 = 1. F 0 1 P 4 P Asmptotes of a Hperbola in Standard Position To find equations for the asmptotes of the hperbola with equation _ a - _ = 1, it helps to eamine the special case when a b and b both equal 1. (This is like eamining the unit circle to learn about ellipses.) Equations for Some Hperbolas 8
Chapter 1 Then - = 1. The hperbola with this equation is smmetric to both aes. Consequentl, each point on the hperbola in the first quadrant has reflection images on the hperbola in other quadrants. The graph at the right shows the reflection images of A, B, C, and D over the -ais and the -ais. A = (1, 0) B = (, ) (, 1.7) ( 4, 15 ) (, 8 ) (, ) ( 1, 0) (, ) (, 8 ) ( 4, 15 ) asmptote = - = 1 (4, 15 )=D (, 8 )=C (, )=B (1, 0) = A (, ) (, 8 ) (4, 15 ) asmptote = C = (, 8 ) (,.8) D = (4, 15 ) (4,.87) The lines = and = appear to be the asmptotes of - = 1. We can verif the equations for the asmptotes algebraicall. When - = 1, = - 1. So = ± - 1. As values of get larger, - 1 becomes closer to, which is. However, because - 1, the curve - = 1 never intersects the lines with equations = or =. So, gets closer to or but never reaches it. According to the Graph Scale-Change Theorem, the scale change S a, b maps - = 1 onto _ a - _ = 1. Under the same scale b change, the asmptotes = ± of - = 1 are mapped onto the lines with equations _ b = ± _ a. These lines are the asmptotes of _ a - _ b = 1. Asmptotes of a Hperbola Theorem The asmptotes of the hperbola with equation _ a - _ b = 1 are _ b = ± a _, or = ± _ b a. QY QY What are the asmptotes of the hperbola in Eample 1? 84 Quadratic Relations
Lesson 1-6 Graphing a Hperbola with Equation in Standard Form To graph _ a - _ = 1 b hand, notice that (a, 0) and b ( a, 0) satisf the equation. These are the vertices of the hperbola. When = 0, is not a real number, so the hperbola does not intersect the -ais. Use the asmptotes to make an accurate sketch of the graph. Remember that the asmptotes are not part of the hperbola. ( a, 0) (a, 0) ( c, 0) (c, 0) b = a b = a Eample Graph the hperbola with equation 16-6 = 1. Solution The equation is in standard form. So, a = 16 and a = 4. The vertices are (4, 0) and ( 4, 0). The asmptotes are _ = 6 ± _ 4, or = ±. Carefull graph the vertices and asmptotes. Then sketch the hperbola. Check Solve _ 16 - _ 6 = 1 for on a CAS. One CAS solution is shown below. 8 ( 4, 0) 6 4 8 6 4 4 6 (4, 0) 4 6 8 = 8 = The complete solution is = - 16 and - 16 0 or = - 16 and 16 0. So = - 16 or = - 16. Graph both equations on the same aes on a graphing utilit. Although the graphing utilit ma have trouble graphing values close to the vertices of the hperbola, the output closel resembles the hand-drawn solution. Equations for Some Hperbolas 85
Chapter 1 Questions COVERING THE IDEAS 1. Fill in the Blanks A hperbola with foci (c, 0) and ( c, 0) and focal constant a has an equation of the form? and vertices at? and?.. Fill in the Blanks A hperbola with equation _ a - _ b = 1 has asmptotes =? and =?. In and 4, an equation for a hperbola is given. Identif its vertices, its foci, and its asmptotes.. 1 = - 4. _ 7 - _ = 1 5. True or False The focal constant of a hperbola equals the distance between the foci. 6. True or False If and are the foci of a hperbola, then F 1 is a line of smmetr for the curve. 7. What does the phrase - 1 is close to for large values of mean? 8. Consider the hperbola with equation _ 5 - _ 64 = 1. a. Name its vertices and state equations for its asmptotes. b. Graph the hperbola. APPLYING THE MATHEMATICS 9. Eplain wh = is not an equation describing the asmptotes of - = 1. 10. Write an equation for the hperbola with vertices at (4, 0) and ( 4, 0) and one focus at (7, 0). 11. The point ( 6, ) is on a hperbola with foci (4, 0) and ( 4, 0). a. Find the focal constant of the hperbola. b. Give an equation for this hperbola in standard form. (Hint: Find b using b = c - a.) c. Graph this hperbola. 1. Show that _ 91 - _ = 1 is equivalent to an equation of the 49 general form A + B + C + D + E + F = 0 b finding the values of A, B, C, D, E, and F. 1. Solve - = 1 for. Use our solution to graph - = 1 on a graphing utilit. 86 Quadratic Relations
Lesson 1-6 REVIEW 14. In Australia, a tpe of football is plaed on elliptical fields. One such field has a major ais of length 185 meters and minor ais of length 155 meters. Surrounding it is an elliptical fence with major ais of length 187 meters and minor ais of length 157 meters. The 1-meter wide track between the fence and the field is to be covered with turf. Find the area of the track. (Lesson 1-5) In 15 and 16, graph the ellipse with the given equation. (Lesson 1-4) 15. _ 4 + _ 5 = 1 16. _ 9 + = 1 17. Standard Quonset huts are semicircular with a diameter of 0 feet and a length of 48 feet. (Lesson 1-) a. Inside the hut, how close to either side of the hut could a 6-foot soldier stand upright? b. What is the volume of a hut? 18. An auto dealer is having a Fourth of Jul etravaganza. The During World War II, eas-tobuild Quonset huts were used dealership plans to be open for 7 hours straight. Suppose the dealer has 100 new cars on the lot and is able to sell an average as barracks for troops. of 4 cars ever hours. (Lesson -1) a. Let h be the number of hours the car dealership has h 0??? been open and let C be the number of cars C 100 96??? remaining on the lot. Find three other pairs of values that satisf this relation and complete the table. b. Write a formula for the number of cars C on the lot as a function of the number of hours h the sale has been on. c. After how man hours will there be onl 60 cars left? d. If the dealership is able to maintain the pace of 4 cars sold ever hours, will the dealer sell all the cars on the lot during the sale? How can ou tell? Field not to scale Fence EXPLORATION 19. The words ellipsis and hperbole have literar meanings. What are these meanings? 0. In Round the Moon, a novel written b Jules Verne in 1870, a group of men launch a rocket to the Moon. During the journe the argue whether the rocket trajector is hperbolic or parabolic. Because each curve is infinite, the men believe the are doomed to travel infinitel through space. Find out on which trajector modern da rockets travel and whether or not the men had reason to worr. = ± _ 4 QY ANSWER Equations for Some Hperbolas 87