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Unformatted text preview: 9.2 THE ELLIPSE AND HYPERBOLA 531 In elliptical rooms called “whispering chambers,” a whisper at one focus, inaudible nearby, is easily heard at the other focus. You will experience this phenomenon if you visit the Statuary Room in the Capitol in Washington, D.C. In many hospitals there are elliptical water tubs designed to break up kidney stones. The patient is positioned so that the stone is at one focus. Small vibrations set off at the other focus are so efﬁciently concentrated that the stone is shattered. Hyperbolic Reﬂectors
A straightforward calculation that you are asked to carry out in the Exercises shows that at each point P of a hyperbola, the tangent line bisects the angle between the focal radii F1 P and F2 P .
hyperbolic reflector F1 (9.2.7) The optical consequences of this are illustrated in Figure 9.2.14. There you see the right branch of a hyperbola with foci F1 , F2 . Light or sound aimed at F2 from any point to the left of the reﬂector is beamed to F1 .
An Application to Range Finding If observers, located at two listening posts at a known distance apart, time the ﬁring of a cannon, the time difference multiplied by the velocity of sound gives the value of 2a and hence determines a hyperbola on which the cannon must be located. A third listening post gives two more hyperbolas. The cannon is found where the hyperbolas intersect. A system for longrange navigation called LORAN is based on a similar use of hyperbolas. F2 Figure 9.2.14 EXERCISES 9.2
Find (a) the center, (b) the foci, (c) the length of the major axis, and (d) the length of the minor axis of the given ellipse. Then sketch the ﬁgure. 1. 3. 5. 7. 8. x2 /9 + y2 /4 = 1. 2. x2 /4 + y2 /9 = 1. 4. 3x2 + 4y2 − 12 = 0. 3x2 + 2y2 = 12. 2 2 6. 4x2 + y2 − 6y + 5 = 0. 4x + 9y − 18y = 27. 4(x − 1)2 + y2 = 64. 16(x − 2)2 + 25( y − 3)2 = 400. 20. 21. 22. 23. 24. foci at (0, −13), (0, 13); transverse axis 24. foci at (−5, 1), (5, 1); transverse axis 6. foci at (−3, 1), (7, 1); transverse axis 6. foci at (−1, −1), (−1, 1); transverse axis 1 . 2 foci at (2, 1), (2, 5); transverse axis 3. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis of the given hyperbola. Then sketch the ﬁgure. 25. 27. 29. 31. 32. 33. x2 − y2 = 1. x2 /9 − y2 /16 = 1. y2 /16 − x2 /9 = 1. (x − 1)2 /9 − ( y − 3)2 /16 = 1. (x − 1)2 /16 − ( y − 3)2 /9 = 1. 4x2 − 8x − y2 + 6y − 1 = 0. 26. y2 − x2 = 1. 28. x2 /16 − y2 /9 = 1. 30. y2 /9 − x2 /16 = 1. Find an equation for the ellipse that satisﬁes the given conditions. 9. 10. 11. 12. 13. 14. 15. 16. foci at (−1, 0), (1, 0); major axis 6. foci at (0, −1), (0, 1); major axis 6. foci at (1, 3), (1, 9); minor axis 8. foci at (3, 1), (9, 1); minor axis 10. focus at (1, 1); center at (1, 3); major axis 10. center at (2, 1); vertices at (2, 6), (1, 1). major axis 10; vertices at (3, 2), (3, −4). focus at (6, 2); vertices at (1, 7), (1, −3). 34. −3x2 +y2 −6x = 0. Find an equation for the indicated hyperbola. 17. foci at (−5, 0), (5, 0); transverse axis 6. 18. foci at (−13, 0), (13, 0); transverse axis 10. 19. foci at (0, −13), (0, 13); transverse axis 10. 35. What is the length of the string in Figure 9.2.1? 36. Show that the set of all points (a cos t , b sin t ) with t real lie on an ellipse. 37. Find the distance between the foci of an ellipse of area A if the length of the major axis is 2a. 38. Show that in an ellipse the product of the distances between the foci and a tangent to the ellipse [d (F1 , T1 ) d (F2 , T2 ) in Figure 9.2.13] is always the square of onehalf the length of the minor axis. 532 CHAPTER 9 THE CONIC SECTIONS; POLAR COORDINATES; PARAMETRIC EQUATIONS 39. Locate the foci of the ellipse given that the point (3, 4) lies on the ellipse and the ends of the major axis are at (0, 0) and (10, 0). 40. Find the centroid of the ﬁrstquadrant portion of the elliptical region b2 x2 + a2 y2 ≤ a2 b2 . 41. Find the center, the vertices, the foci, the asymptotes, and the length of the transverse axis of the hyperbola with equation xy = 1. HINT: Deﬁne new XY coordinates by setting x = X + Y and y = X − Y . For Exercises 42–44 we refer to the hyperbola in Figure 9.2.8. 42. Find functions x = x(t ), y = y(t ) such that, as t ranges over the set of real numbers, the points (x(t ), y(t )) traverse (a) the right branch of the hyperbola. (b) the left branch of the hyperbola. 43. Find the area of the region between the right branch of the hyperbola and the vertical line x = 2a. 44. Show that at each point P of the hyperbola the tangent at P bisects the angle between the focal radii F1 P and F2 P . Although all parabolas have exactly the same shape (Exercise 41, Section 9.1), ellipses come in different shapes. The shape of an ellipse depends on its eccentricity e. This is half the distance between the foci divided by half the length of the major axis:
(9.2.8) The shape of a hyperbola is determined by its eccentricity e. This is half the distance between the foci divided by half the length of the transverse axis:
(9.2.9) e = c/a. For all hyperbolas, e > 1. In Exercises 55–58, hyperbola. 55. x2 /9 − y2 /16 = 1. 56. x2 /16 − y2 /9 = 1. 57. x2 − y2 = 1. 58. x2 /25 − y2 /144 = 1. 59. Suppose H1 and H2 are both hyperbolas with the same transverse axis. Compare the shape of H1 to the shape of H2 if e1 < e2 . 60. What happens to a hyperbola if e tends to 1? 61. What happens to a hyperbola if e increases without bound? 62. (Compare to Exercise 54.) Let l be a line and let F be a point not on l . Show that, if e > 1, then the set of all points P for which d (F , P ) = e d (l , P ) is a hyperbola of eccentricity, e. HINT: Begin by choosing a coordinate system whereby F falls on the origin and l is a vertical line x = k . c 63. A meteor crashes somewhere in the hills that lie north of point A. The impact is heard at point A and four seconds later it is heard at point B. Two seconds still later it is heard at point C . Locate the point of impact given that A lies two miles due east of B and two miles due west of C . (Take 0.20 miles per second as the speed of sound.) determine the eccentricity of the e = c/a. For every ellipse, 0 < e < 1. In Exercises 45–48, determine the eccentricity of the ellipse. 46. x2 /16 + y2 /25 = 1. x2 /25 + y2 /16 = 1. (x − 1)2 /25 + ( y + 2)2 /9 = 1. (x + 1)2 /169 + ( y − 1)2 /144 = 1. Suppose that E1 and E2 are both ellipses with the same major axis. Compare the shape of E1 to the shape of E2 if e1 < e2 . 50. What happens to an ellipse with major axis 2a if e tends to 0? 51. What happens to an ellipse with major axis 2a if e tends to 1? In Exercises 52 and 53, write an equation for the ellipse. 45. 47. 48. 49. 52. Major axis from (−3, 0) to (3, 0), eccentricity 1 . 3 √ 2 53. Major axis from (−3, 0) to (3, 0), eccentricity 3 2. 54. Let l be a line and let F be a point not on l . You have seen that the set of points P for which d (F , P ) = d (l , P ) is a parabola. Show that, if 0 < e < 1, then the set of all points P for which d (F , P ) = e d (l , P ) is an ellipse of eccentricity e. HINT: Begin by choosing a coordinate system whereby F falls on the origin and l is a vertical line x = k . c 64. A radio signal is received at the points marked P1 , P2 , P3 , P4 in the ﬁgure. Suppose that the signal arrives at P1 600 microseconds after it arrives at P2 and arrives at P4 800 microseconds after it arrives at P3 . Locate the source of the signal given that radio waves travel at the speed of light, 186,000 miles per second. (A microsecond is a millionth of a second.)
y miles P3 100 P1 100 100 100 P4 P2 x miles ...
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