1031 10.4 Summary of the Conic Sections Find the eccentricity e of each conic section. The point shown on the x-axis is a focus, and the line shown is a directrix. 45. x x = 27 (–3, 8) (3, 0) y 46. x y x = –9 (–4, 0) (4, ) 10 3 47. x y (Ë2 , 0) x = –Ë2 48. (27, 0) x y x = 4 (–27, 48 )3 4 49. x (9, 0) (9, –7.5) x = 4 y 50. x (20, 0) (5, 20) x = –20 y 37. x2 = 4y - 8 38. 1x - 422 8 + 1y + 122 2 = 0 39. y2 - 4y = x + 4 40. 1x + 722 + 1y - 522 + 4 = 0 41. 3x2 + 6x + 3y2 - 12y = 12 42. -4x2 + 8x + y2 + 6y = -6 43. 4x2 - 8x + 9y2 - 36y = -4 44. 3x2 + 12x + 3y2 = 0 Satellite Trajectory When a satellite is near Earth, its orbital trajectory may trace out a hyperbola, a parabola, or an ellipse. The type of trajectory depends on the satellite’s velocity V in meters per second. It will be hyperbolic if V7 k2 D , parabolic if V = k2 D , or elliptical if V6 k2 D , where k = 2.82 * 107 is a constant and D is the distance in meters from the satellite to the center of Earth. (Data from Loh, W., Dynamics and Thermodynamics of Planetary Entry, Prentice-Hall, and Thomson, W., Introduction to Space Dynamics, John Wiley and Sons.) 51. When the artificial satellite Explorer IV was at a maximum distance D of 42.5 * 106 m from Earth’s center, it had a velocity V of 2090 m per sec. Determine the shape of its trajectory. 52. If a satellite is scheduled to leave Earth’s gravitational influence, its velocity must be increased so that its trajectory changes from elliptical to hyperbolic. Determine the minimum increase in velocity necessary for Explorer IV to escape Earth’s gravitational influence when D= 42.5 * 106 m. Round to the nearest whole number.
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