SECTION 10.5 Rotation of Axes; General Form of a Conic 727 5. To transform the equation + + + + + = ≠ Ax Bxy Cy Dx Ey F B 0 0 2 2 into one in ′x and ′y without an ′ ′ x y-term, rotate the axes through an acute angle θ that satisfies the equation . 6. Multiple Choice Except for degenerate cases, the equation + + + + + = Ax Bxy Cy Dx Ey F 0 2 2 defines a(n) if − = B AC 4 0. 2 (a) circle (b) ellipse (c) hyperbola (d) parabola Concepts and Vocabulary 7. Except for degenerate cases, the equation + + + + + = Ax Bxy Cy Dx Ey F 0 2 2 defines an ellipse if . 8. Multiple Choice The equation + + = ax y y 6 12 0 2 2 defines an ellipse if . (a) < a 0 (b) =a 0 (c) > a 0 (d) a is any real number 9. True or False The equation + + = x Bxy y 3 12 10 2 2 defines a parabola if = − B 12. 10. True or False To eliminate the xy-term from the equation − + − + + = x xy y x y 2 2 3 5 0, 2 2 rotate the axes through an angle θ, where θ = − B AC cot 4 . 2 Skill Building In Problems 11–20, identify the graph of each equation without completing the squares. 11. + + + = x x y 4 3 0 2 12. − + = y y x 2 3 3 0 2 13. + − + = x y x y 6 3 12 6 0 2 2 14. + − + + = x y x y 2 8 4 2 0 2 2 15. − + + = x y x 3 2 6 4 0 2 2 16. − − + + = x y x y 4 3 8 6 1 0 2 2 17. − − + = y x y x 2 0 2 2 18. − − − = y x x y 8 2 0 2 2 19. + − + = x y x y 8 4 0 2 2 20. + − + = x y x y 2 2 8 8 0 2 2 In Problems 21–30, determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 21. + + − = x xy y 4 3 0 2 2 22. − + − = x xy y 4 3 0 2 2 23. + + − = x xy y 5 6 5 8 0 2 2 24. − + − = x xy y 3 10 3 32 0 2 2 25. − + − = x xy y 13 6 3 7 16 0 2 2 26. + + − = x xy y 11 10 3 4 0 2 2 27. − + − − = x xy y x y 4 4 8 5 16 5 0 2 2 28. + + + + = x xy y y 4 4 5 5 5 0 2 2 29. − + − − = x xy y x y 25 36 40 12 13 8 13 0 2 2 30. − + − = x xy y 34 24 41 25 0 2 2 In Problems 31–42, rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation. Refer to Problems 21–30 for Problems 31–40. 31. + + − = x xy y 4 3 0 2 2 32. − + − = x xy y 4 3 0 2 2 33. + + − = x xy y 5 6 5 8 0 2 2 34. − + − = x xy y 3 10 3 32 0 2 2 35. − + − = x xy y 13 6 3 7 16 0 2 2 36. + + − = x xy y 11 10 3 4 0 2 2 37. − + − − = x xy y x y 4 4 8 5 16 5 0 2 2 38. + + + + = x xy y y 4 4 5 5 5 0 2 2 39. − + − − = x xy y x y 25 36 40 12 13 8 13 0 2 2 40. − + − = x xy y 34 24 41 25 0 2 2 41. + + − + = x xy y x y 16 24 9 130 90 0 2 2 42. + + − + = x xy y x y 16 24 9 60 80 0 2 2 In Problems 43–52, identify the graph of each equation without applying a rotation of axes. 43. + − + + + = x xy y x y 3 2 3 2 5 0 2 2 44. − + + + − = x xy y x y 2 3 4 2 3 5 0 2 2 45. − + − − = x xy y y 7 3 10 0 2 2 46. − + − − = x xy y x 2 3 2 4 2 0 2 2 47. + + − − = x xy y x y 9 12 4 0 2 2 48. + + − − + = x xy y x y 10 12 4 10 0 2 2 49. − + − − − = x xy y x y 10 12 4 10 0 2 2 50. + + − − = x xy y x y 4 12 9 0 2 2 51. − + + + − = x xy y x y 3 2 4 2 1 0 2 2 52. + + + − + = x xy y x y 3 2 4 2 10 0 2 2 53. Satellite Receiver A parabolic satellite receiver is initially positioned so its axis of symmetry is parallel to the x-axis. A motor allows the receiver to rotate and track the satellite signal. If the rotated receiver has the equation − − + − + = x xy x y y 4 4 21 21 21 2 21 171 324 0 2 2 through what acute angle did the receiver rotate, to the nearest tenth of a degree? 54. Elliptical Trainer A runner on an elliptical trainer inclines the machine to increase the difficulty of her workout. If the initial elliptic path of the pedals had a horizontal major axis, and the inclined path has the equation − − + − + = x xy x y y 20 10 19 2 89 89 48 5 0 2 2 through what acute angle did the runner incline the machine, to the nearest tenth of a degree? Applications and Extensions
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