SECTION 8.1 Parabolas Chapter 8 Analytic Geometry Section 8.1: Parabolas Equations of Parabolas Equations of Parabolas Definition of a Parabola: Equations of Parabolas with Vertex at the Origin: MATH 1330 Precalculus 635 CHAPTER 8 Analytic Geometry 636 University of Houston Department of Mathematics SECTION 8.1 Parabolas Example: Solution: MATH 1330 Precalculus 637 CHAPTER 8 Analytic Geometry Additional Equations of Parabolas with Vertex at the Origin: Opening Downward: 638 University of Houston Department of Mathematics SECTION 8.1 Parabolas Opening to the Right: Opening to the Left: Example: MATH 1330 Precalculus 639 CHAPTER 8 Analytic Geometry Solution: The Standard Form for the Equation of a Parabola: 640 University of Houston Department of Mathematics SECTION 8.1 Parabolas Opening Upward: Example: MATH 1330 Precalculus 641 CHAPTER 8 Analytic Geometry Solution: Opening Downward: 642 University of Houston Department of Mathematics SECTION 8.1 Parabolas Opening to the Right: MATH 1330 Precalculus 643 CHAPTER 8 Analytic Geometry Opening to the Left: 644 University of Houston Department of Mathematics SECTION 8.1 Parabolas Example: Solution: MATH 1330 Precalculus 645 CHAPTER 8 Analytic Geometry Example: Solution: 646 University of Houston Department of Mathematics SECTION 8.1 Parabolas Tangents to Parabolas: Example: MATH 1330 Precalculus 647 CHAPTER 8 Analytic Geometry Solution: Additional Example 1: Solution: 648 University of Houston Department of Mathematics SECTION 8.1 Parabolas Additional Example 2: Solution: MATH 1330 Precalculus 649 CHAPTER 8 Analytic Geometry Additional Example 3: Solution: 650 University of Houston Department of Mathematics SECTION 8.1 Parabolas Additional Example 4: Solution: MATH 1330 Precalculus 651 CHAPTER 8 Analytic Geometry Additional Example 5: Solution: 652 University of Houston Department of Mathematics SECTION 8.1 Parabolas Additional Example 6: Solution: MATH 1330 Precalculus 653 CHAPTER 8 Analytic Geometry 654 University of Houston Department of Mathematics Exercise Set 8.1: Parabolas Write each of the following equations in the standard 13. xy22 8 5 form for the equation of a parabola, where the standard form is represented by one of the following 2 14. yx4 16 equations: 2 2 2 x h 4 p y k x h 4 p y k 15. yx43 2 2 y k 4 p x h y k 4 p x h 16. xy3412 17. (xy 5)2 2 4 1. y2 14 y 2 x 43 0 2 2. x2 10 x 12 y 61 18. yx1 10 3 2 3. 9y x2 8 x 10 19. yx6 1 2 4. 7x y2 10 y 24 20. xy12 8 6 2 5. x3 y 24 y 50 21. x2 12 x 6 y 24 0 2 6. y2 x 12 x 15 22. x2 2 y 8 x 10 2 7. 3x 3 x 5 y 0 23. y2 8 x 4 y 36 2 8. 5y 5 y x 6 24. y2 6 y 4 x 5 0 2 25. x25 16 y 10 x For each of the following parabolas, 26. y2 10 y x 28 0 (a) Write the given equation in the standard form for the equation of a parabola. (Some 27. y2 4 y 2 x 4 0 equations may already be given in standard form.) 28. 12y x2 4 x 16 It may be helpful to begin sketching the graph for 2 part (h) as a visual aid to answer the questions 29. 3y 30 y 8 x 67 0 below. 30. 5x2 30 x 16 y 19 (b) State the equation of the axis. 2 (c) State the coordinates of the vertex. 31. 2x 8 x 7 y 34 (d) State the equation of the directrix. 32. 4y2 8 y 9 x 40 0 (e) State the coordinates of the focus. (f) State the focal width. (g) State the coordinates of the endpoints of the Use the given features of each of the following focal chord. parabolas to write an equation for the parabola in (h) Sketch a graph of the parabola which includes standard form. the features from (c)-(e) and (g). Label the vertex V and the focus F. 33. Vertex: 2, 5 Focus: 4, 5 2 9. xy40 34. Vertex: 1, 3 10. yx2 12 0 Focus: 1, 0 2 11. 10xy 12. 6yx2 MATH 1330 Precalculus 655 Exercise Set 8.1: Parabolas 35. Vertex: 2, 0 48. Endpoints of focal chord: 2, 3 and 6, 3 Focus: 2, 4 Opens downward 36. Vertex: 4, 2 Focus: 6, 2 Answer the following. 49. Write an equation of the line tangent to the 37. Focus: 2, 3 parabola with equation f x x2 54 x at: Directrix: y 9 (a) x 3 (b) x 2 38. Focus: 4,1 Directrix: y 5 50. Write an equation of the line tangent to the parabola with equation f x 3 x2 6 x 1 at: 39. Focus: 4, 1 (a) x 0 1 Directrix: x 7 2 (b) x 1 40. Focus: 3, 5 51. Write an equation of the line tangent to the 2 Directrix: x 4 parabola with equation f x 2 x 5 x 1 at: (a) x 1 41. Focus: 4, 2 1 (b) x Opens downward 2 p 7 52. Write an equation of the line tangent to the 42. Focus: 1, 5 parabola with equation f x x2 42 x at: Opens to the right (a) x 3 p 3 3 (b) x 2 43. Vertex: 5, 6 Opens upward 53. Write an equation of the line tangent to the Length of focal chord: 6 parabola with equation f x 4 x2 5 x 3 at the point 1, 4 . 1 44. Vertex: 0, 2 Opens to the left 54. Write an equation of the line tangent to the Length of focal chord: 2 parabola with equation f x 2 x2 5 x 4 at the point 3, 7 . 45. Vertex: 3, 2 Horizontal axis 55. Write an equation of the line tangent to the Passes through 6, 5 parabola with equation f x x2 61 x at 46. Vertex: 2,1 the point 5, 6 . Vertical axis Passes through 8, 5 56. Write an equation of the line tangent to the 2 parabola with equation f x 3 x 4 x 9 at 47. Endpoints of focal chord: 0, 5 and 0, 5 the point 2, 5 . Opens to the left 656 University of Houston Department of Mathematics Exercise Set 8.1: Parabolas Give the point(s) of intersection of the parabola and the line whose equations are given. 57. f x x2 4 x 11 g x 53 x 58. f x x2 81 x g x 2 x 10 59. f x x2 10 x 10 g x 89 x 60. f x x2 5 x 10 g x 7 x 14 61. f x x2 65 x g x 6 x 14 62. f x x2 32 x g x 5 x 13 63. f x 3 x2 6 x 1 g x 37 x 64. f x 2 x2 8 x 5 g x 65 x MATH 1330 Precalculus 657 CHAPTER 8 Analytic Geometry Section 8.2: Ellipses Equations of Ellipses Equations of Ellipses Definition of an Ellipse: Equations of Ellipses with Center at the Origin: 658 University of Houston Department of Mathematics SECTION 8.2 Ellipses MATH 1330 Precalculus 659 CHAPTER 8 Analytic Geometry 660 University of Houston Department of Mathematics SECTION 8.2 Ellipses Example: Solution: MATH 1330 Precalculus 661 CHAPTER 8 Analytic Geometry Example: 662 University of Houston Department of Mathematics SECTION 8.2 Ellipses Solution: The Standard Form for the Equation of an Ellipse: MATH 1330 Precalculus 663 CHAPTER 8 Analytic Geometry 664 University of Houston Department of Mathematics SECTION 8.2 Ellipses Example: Solution: MATH 1330 Precalculus 665 CHAPTER 8 Analytic Geometry Example: 666 University of Houston Department of Mathematics SECTION 8.2 Ellipses Solution: Special Case: MATH 1330 Precalculus 667 CHAPTER 8 Analytic Geometry Example: Solution: Additional Example 1: Solution: 668 University of Houston Department of Mathematics SECTION 8.2 Ellipses Additional Example 2: Solution: MATH 1330 Precalculus 669 CHAPTER 8 Analytic Geometry Additional Example 3: Solution: 670 University of Houston Department of Mathematics SECTION 8.2 Ellipses Additional Example 4: Solution: MATH 1330 Precalculus 671 CHAPTER 8 Analytic Geometry 672 University of Houston Department of Mathematics Exercise Set 8.2: Ellipses Write each of the following equations in the standard (e) State the coordinates of the foci. form for the equation of an ellipse, where the standard (f) State the eccentricity. form is represented by one of the following equations: (g) Sketch a graph of the ellipse which includes 22 22 x h y k x h y k the features from (b)-(e). Label the center C, 221 221 and the foci F1 and F2. ab ba xy22 22 11. 1 1. 25xy 4 100 0 9 49 22 2. 9xy 16 144 0 xy22 12. 1 22 36 4 3. 9x 36 x 4 y 32 y 64 0 2 2 22 x 2 y 4. 4x 24 x 16 y 32 y 12 0 13. 1 16 4 5. 3x22 2 y 30 x 12 y 87 2 x2 y 1 22 14. 1 6. x8 y 113 14 x 48 y 95 22 7. 16x 16 x 64 8 y 24 y 42 xy2322 15. 1 25 16 8. 18x22 9 y 153 24 x 6 y xy5222 16. 1 Answer the following. 16 25 9. (a) What is the equation for the eccentricity, e, xy4322 of an ellipse? 17. 1 (b) As e approaches 1, the ellipse appears to 91 become more (choose one): 22 xy23 elongated circular 18.