616 9 Additional Topics in Analtic Geometr 53. Space Science. A designer of a 00-foot-diameter parabolic electromagnetic antenna for tracking space probes wants to place the focus 100 feet above the verte (see the figure). (A) ind the equation of the parabola using the ais of the parabola as the ais (up positive) and verte at the origin. (B) Determine the depth of the parabolic reflector. 54. Signal Light. A signal light on a ship is a spotlight with parallel reflected light ras (see the figure). Suppose the parabolic reflector is 1 inches in diameter and the light source is located at the focus, which is 1.5 inches from the verte. 00 ft Signal light ocus ocus Radiotelescope 100 ft (A) ind the equation of the parabola using the ais of the parabola as the ais (right positive) and verte at the origin. (B) Determine the depth of the parabolic reflector. SECTION 9- Ellipse Definition of an Ellipse Drawing an Ellipse Standard Equations and Their Graphs Applications We start our discussion of the ellipse with a coordinate-free definition. Using this definition, we show how an ellipse can be drawn and we derive standard equations for ellipses speciall located in a rectangular coordinate sstem. Definition of an Ellipse The following is a coordinate-free definition of an ellipse: DEINITION 1 Ellipse An ellipse is the set of all points P in a plane such that the sum of the distances of P from two fied points in the plane is constant. Each of the fied points, and, is called a focus, and together the are called foci. Referring to the figure, the line segment VV through the foci is the major ais. The perpendicular bisector BB of the major ais is the minor ais. Each end of the major ais,
9- Ellipse 617 V and V, is called a verte. The midpoint of the line segment is called the center of the ellipse. V d 1 d Constant B d 1 P d B V Drawing an Ellipse An ellipse is eas to draw. All ou need is a piece of string, two thumbtacks, and a pencil or pen (see ig. 1). Place the two thumbtacks in a piece of cardboard. These form the foci of the ellipse. Take a piece of string longer than the distance between the two thumbtacks this represents the constant in the definition and tie each end to a thumbtack. inall, catch the tip of a pencil under the string and move it while keeping the string taut. The resulting figure is b definition an ellipse. Ellipses of different shapes result, depending on the placement of thumbtacks and the length of the string joining them. IGURE 1 Drawing an ellipse. Note that d 1 d alwas adds up to the length of the string, which does not change. P d 1 d ocus String ocus Standard Equations and Their Graphs Using the definition of an ellipse and the distance-between-two-points formula, we can derive standard equations for an ellipse located in a rectangular coordinate sstem. We start b placing an ellipse in the coordinate sstem with the foci on the ais equidistant from the origin at (c, 0) and (c, 0), as in igure. IGURE Ellipse with foci on ais. P(, ) d 1 d (c, 0) 0 (c, 0) d 1 d Constant c 0
618 9 Additional Topics in Analtic Geometr or reasons that will become clear soon, it is convenient to represent the constant sum d 1 d b a, a 0. Also, the geometric fact that the sum of the lengths of an two sides of a triangle must be greater than the third side can be applied to igure to derive the following useful result: d(, P) d(p, ) d(, ) d 1 d c a c a c (1) We will use this result in the derivation of the equation of an ellipse, which we now begin. Referring to igure, the point P(, ) is on the ellipse if and onl if d 1 d a d(p, ) d(p, ) a ( c) ( 0) ( c) ( 0) a After eliminating radicals and simplifing, a good eercise for ou, we obtain (a c ) a a (a c ) a a c 1 () (3) Dividing both sides of equation () b a (a c ) is permitted, since neither a nor a c is 0. rom equation (1), a c; thus a c and a c 0. The constant a was chosen positive at the beginning. To simplif equation (3) further, we let to obtain b a c b 0 (4) a b 1 (5) rom equation (5) we see that the intercepts are a and the intercepts are b. The intercepts are also the vertices. Thus, Major ais length a Minor ais length b To see that the major ais is longer than the minor ais, we show that a b. Returning to equation (4), b a c a, b, c 0 b c a
9- Ellipse 619 b a Definition of b a 0 (b a)(b a) 0 b a 0 Since b a is positive, b a must be negative. b a b a a b Length of major ais Length of minor ais If we start with the foci on the ais at (0, c) and (0, c) as in igure 3, instead of on the ais as in igure, then, following arguments similar to those used for the first derivation, we obtain b a 1 a b (6) where the relationship among a, b, and c remains the same as before: b a c (7) The center is still at the origin, but the major ais is now along the ais and the minor ais is along the ais. IGURE 3 Ellipse with foci on ais. (0, c) d 1 P(, ) 0 d (0, c) d 1 d Constant c 0 To sketch graphs of equations of the form (5) or (6) is an eas matter. We find the and intercepts and sketch in an appropriate ellipse. Since replacing with, or with produces an equivalent equation, we conclude that the graphs are smmetric with respect to the ais, ais, and origin. If further accurac is required, additional points can be found with the aid of a calculator and the use of smmetr properties. Given an equation of the form (5) or (6), how can we find the coordinates of the foci without memorizing or looking up the relation b a c? There is a simple geometric relationship in an ellipse that enables us to get the same result using the Pthagorean theorem. To see this relationship, refer to igure 4(a). Then, using the
60 9 Additional Topics in Analtic Geometr definition of an ellipse and a for the constant sum, as we did in deriving the standard equations, we see that Thus: d d a d a d a The length of the line segment from the end of a minor ais to a focus is the same as half the length of a major ais. This geometric relationship is illustrated in igure 4(b). Using the Pthagorean theorem for the triangle in igure 4(b), we have or b c a b a c Equations (4) and (7) or c a b Useful for finding the foci, given a and b Thus, we can find the foci of an ellipse given the intercepts a and b simpl b using the triangle in igure 4(b) and the Pthagorean theorem. IGURE 4 Geometric relationships. b a b 1 a b 0 b a b c a d c 0 c a d a c 0 c a a b b (a) (b) We summarize all of these results for convenient reference in Theorem 1. Theorem 1 Standard Equations of an Ellipse with Center at (0, 0) 1. a 1 a b 0 b
9- Ellipse 61 intercepts: a (vertices) intercepts: b oci: (c, 0), (c, 0) c a b Major ais length a Minor ais length b a b a c 0 c a b. b 1 a b 0 a intercepts: b intercepts: a (vertices) oci: (0, c), (0, c) a c a c a b Major ais length a Minor ais length b b 0 b [Note: Both graphs are smmetric with respect to the ais, ais, and origin. Also, the major ais is alwas longer than the minor ais.] c a EXPLORE-DISCUSS 1 The line through a focus of an ellipse that is perpendicular to the major ais intersects the ellipse in two points G and H. or each of the two standard equations of an ellipse with center (0, 0), find an epression in terms of a and b for the distance from G to H. EXAMPLE 1 Graphing Ellipses Sketch the graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor aes. (A) 9 16 144 (B) 10 Solutions (A) irst, write the equation in standard form b dividing both sides b 144: 9 16 144 @@@@@@@@e? @@h? @@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@ 9 16 144 144 144 144 @@ @@ @@g @@@@@@@@?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?@@@@@@@@ 16 9 1 a 16 and b 9
6 9 Additional Topics in Analtic Geometr Locate the intercepts: intercepts: intercepts: 4 3 3 4 4 c 0 c 4 3 IGURE 5 9 16 144. and sketch in the ellipse, as shown in igure 5. oci: c a b 16 9 7 c 7 c is positive Thus, the foci are ( 7, 0) and ( 7, 0). Major ais length (4) 8 Minor ais length (3) 6 (B) Write the equation in standard form b dividing both sides b 10: 10 @@@@@@@@e? @@h? @@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@ 10 10 10 10 @@ @@ @@g @@@@@@@@?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@e?@@@@@@@@?e@@@@@@@@?@@@@@@@@ 5 10 1 a 10 and b 5 Locate the intercepts: intercepts: intercepts: 5.4 10 3.16 and sketch in the ellipse, as shown in igure 6. 10 c 10 oci: c a b 10 5 5 0 5 5 c c 5 10 IGURE 6 10. Thus, the foci are (0, 5) and (0, 5). Major ais length 10 6.3 Minor ais length 5 4.47 Remark. To graph the equation 9 16 144 of Eample 1A on a graphing 9 utilit we first solve the equation for, obtaining. We then graph 14416
9- Ellipse 63 each of the two functions. The graph of ellipse, and the graph of 9 14416 9 14416 is the lower half. is the upper half of the Matched Problem 1 Sketch the graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor aes. (A) 4 4 (B) 3 18 EXAMPLE inding the Equation of an Ellipse ind an equation of an ellipse in the form M N 1 M, N 0 10 Solutions if the center is at the origin, the major ais is along the ais, and: (A) Length of major ais 0 (B) Length of major ais 10 Length of minor ais 1 Distance of foci from center 4 (A) Compute and intercepts and make a rough sketch of the ellipse, as shown in igure 7. 10 IGURE 7 1. 36 100 b 10 IGURE 8. 9 5 1 0 5 4 5 5 b 10 a 0 (B) Make a rough sketch of the ellipse, as shown in igure 8; locate the foci and intercepts, then determine the intercepts using the special triangle relationship discussed earlier. b a 1 a 10 5 b 5 4 5 16 9 9 5 1 b a 1 36 100 1 10 b 1 6 b 3
64 9 Additional Topics in Analtic Geometr Matched Problem ind an equation of an ellipse in the form M N 1 M, N 0 if the center is at the origin, the major ais is along the ais, and: (A) Length of major ais 50 (B) Length of minor ais 16 Length of minor ais 30 Distance of foci from center 6 EXPLORE-DISCUSS Consider the graph of an equation in the variables and. The equation of its magnification b a factor k 0 is obtained b replacing and in the equation b /k and /k, respectivel. (A) ind the equation of the magnification b a factor 3 of the ellipse with equation ( /4) 1. Graph both equations. (B) Give an eample of an ellipse with center (0, 0) with a b that is not a magnification of ( /4) 1. (C) ind the equations of all ellipses that are magnifications of ( /4) 1. Applications You are no doubt aware of man occurrences and uses of elliptical forms: orbits of satellites, planets, and comets; shapes of galaies; gears and cams; some airplane wings, boat keels, and rudders; tabletops; public fountains; and domes in buildings are a few eamples (see ig. 9). A fairl recent application in medicine is the use of elliptical reflectors and ultrasound to break up kidne stones. Planet Sun Planetar motion (a) (b) (c) IGURE 9 Uses of elliptical forms. Elliptical gears Elliptical dome Johannes Kepler (1571 1630), a German astronomer, discovered that planets move in elliptical orbits, with the sun at a focus, and not in circular orbits as had been thought before [ig. 9(a)]. igure 9(b) shows a pair of elliptical gears with pivot points at foci. Such gears transfer constant rotational speed to variable rotational speed, and
9- Ellipse 65 vice versa. igure 9(c) shows an elliptical dome. An interesting propert of such a dome is that a sound or light source at one focus will reflect off the dome and pass through the other focus. One of the chambers in the Capitol Building in Washington, D.C., has such a dome, and is referred to as a whispering room because a whispered sound at one focus can be easil heard at the other focus. Answers to Matched Problems 1. (A) oci: (3, 0), (3, 0) Major ais length 4 1 Minor ais length 0 1 (B) 18 oci: (0, 1), (0, 1) Major ais length 18 8.49 Minor ais length 6 4.90 6 6 18. (A) (B) 65 5 1 100 64 1 EXERCISE 9- A In Problems 1 6, sketch a graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor aes. 1.. 5 4 1 3. 4. 4 5 1 4 9 1 5. 9 9 6. 4 4 B 9 4 1 In Problems 7 1, sketch a graph of each equation, find the coordinates of the foci, and find the lengths of the major and minor aes. 7. 5 9 5 8. 16 5 400 9. 1 10. 4 3 4 11. 4 7 8 1. 3 4 In Problems 13 18, find an equation of an ellipse in the form 1 M, N 0 M N if the center is at the origin, and: 13. Major ais on ais Major ais length 6 Minor ais length 14. Major ais on ais Major ais length 3 Minor ais length 30 15. Major ais on ais Minor ais length 10 Distance of foci from center 4 16. Major ais on ais Major ais length 16 Distance of foci from center 7
66 9 Additional Topics in Analtic Geometr 17. Major ais on ais Major ais length 4 Distance between foci 18. Major ais on ais Major ais length 4 Distance between foci 50 19. Eplain wh an equation whose graph is an ellipse does not define a function. 0. Consider all ellipses having (0, 1) as the ends of the minor ais. Describe the connection between the elongation of the ellipse and the distance from a focus to the origin. In Problems 1 4, graph each sstem of equations in the same rectangular coordinate sstem and find the coordinates of an points of intersection. ind noninteger coordinates to three decimal places. Check Problems 1 4 with a graphing utilit.* 1. 16 5 400. 5 16 400 5 10 5 8 0 3. 5 16 400 4. 16 5 400 5 36 0 3 0 0 In Problems 35 38, use a graphing utilit to find the coordinates of all points of intersection to two decimal places. 35. 3 0, 4 5 11 36. 8 35 3,600, 5 37. 50 4 1,05, 9 300 38. 7 95, 13 6 63 APPLICATIONS 39. Engineering. The semielliptical arch in the concrete bridge in the figure must have a clearance of 1 feet above the water and span a distance of 40 feet. ind the equation of the ellipse after inserting a coordinate sstem with the center of the ellipse at the origin and the major ais on the ais. The ais points up, and the ais points to the right. How much clearance above the water is there 5 feet from the bank? In Problems 5 8, find the first-quadrant points of intersection for each sstem of equations to three decimal places. Check Problems 5 8 with a graphing utilit. 5. 5 63 6. 3 4 57 0 0 7. 3 33 8. 3 43 8 0 1 0 In Problems 9 3, determine whether the statement is true or false. If true, eplain wh. If false, give a countereample. 9. The line segment joining the foci of an ellipse has greater length than the minor ais. 30. There is eactl one ellipse with center (0, 0) and foci (1, 0). 31. Ever line through the center of 4 16 intersects the ellipse in eactl two points. 3. Ever nonvertical line through a verte of 4 16 intersects the ellipse in eactl two points. C 33. ind an equation of the set of points in a plane, each of whose distance from (, 0) is one-half its distance from the line 8. Identif the geometric figure. 34. ind an equation of the set of points in a plane, each of whose distance from (0, 9) is three-fourths its distance from the line 16. Identif the geometric figure. *Please note that use of a graphing utilit is not required to complete these eercises. Checking them with a g.u. is optional. 40. Design. A4 8 foot elliptical tabletop is to be cut out of a 4 8 foot rectangular sheet of teak plwood (see the figure). To draw the ellipse on the plwood, how far should the foci be located from each edge and how long a piece of string must be fastened to each focus to produce the ellipse (see igure 1 in the tet)? Compute the answer to two decimal places. Elliptical bridge Elliptical table String 41. Aeronautical Engineering. Of all possible wing shapes, it has been determined that the one with the least drag along the trailing edge is an ellipse. The leading edge ma be a straight line, as shown in the figure. One of the most famous planes with this design was the World War II British Spitfire. The plane in the figure has a wingspan of 48.0 feet.
9-3 Hperbola 67 Leading edge and 1.00 foot in front of it. The chord is 1.00 foot shorter than the major ais. Elliptical wings and tail uselage Trailing edge (A) If the straight-line leading edge is parallel to the major ais of the ellipse and is 1.14 feet in front of it, and if the leading edge is 46.0 feet long (including the width of the fuselage), find the equation of the ellipse. Let the ais lie along the major ais (positive right), and let the ais lie along the minor ais (positive forward). (B) How wide is the wing in the center of the fuselage (assuming the wing passes through the fuselage)? Compute quantities to 3 significant digits. 4. Naval Architecture. Currentl, man high-performance racing sailboats use elliptical keels, rudders, and main sails for the same reasons stated in Problem 41 less drag along the trailing edge. In the accompaning figure, the ellipse containing the keel has a 1.0-foot major ais. The straightline leading edge is parallel to the major ais of the ellipse Rudder Keel (A) ind the equation of the ellipse. Let the ais lie along the minor ais of the ellipse, and let the ais lie along the major ais, both with positive direction upward. (B) What is the width of the keel, measured perpendicular to the major ais, 1 foot up the major ais from the bottom end of the keel? Compute quantities to 3 significant digits. SECTION 9-3 Hperbola Definition of a Hperbola Drawing a Hperbola Standard Equations and Their Graphs Applications As before, we start with a coordinate-free definition of a hperbola. Using this definition, we show how a hperbola can be drawn and we derive standard equations for hperbolas speciall located in a rectangular coordinate sstem. Definition of a Hperbola The following is a coordinate-free definition of a hperbola: