Exercise Set 8.1: Parabolas Write each of the following equations in the standard form for the equation of a parabola, where the standard form is represented by one of the following equations: 13. ( x 2 )2 = 8 ( y + 5 ) 14. ( y 4 )2 = 16 x ( x h) 2 = 4 p( y k ) ( x h) 2 = 4 p ( y k ) 15. y 2 = 4 ( x 3) ( y k) 2 = 4 p ( x h) ( y k) 2 = 4 p ( x h ) 16. ( x + 3)2 = 4 ( y 1) 1. y 2 14 y 2 x + 43 = 0 17. ( x + 5) 2 = 2 ( y 4 ) 2. x 2 + 10 x 12 y = 61 18. ( y + 1)2 = 10 ( x + 3) 3. 9 y = x 2 8 x 10 19. ( y 6 )2 = 1( x 2 ) 4. 7 x = y 2 10 y + 24 20. ( x 1)2 = 8 ( y 6 ) 5. x = 3 y 2 24 y + 50 21. x 2 + 12 x 6 y + 24 = 0 6. y = 2 x 2 + 12 x + 15 22. x 2 2 y = 8 x 10 7. 3x 2 3x 5 y = 0 23. y 2 8 x = 4 y + 36 8. 5 y2 + 5 y = x 6 24. y 2 + 6 y 4 x + 5 = 0 25. x 2 + 25 = 16 y 10 x For each of the following parabolas, (a) Write the given equation in the standard form for the equation of a parabola. (Some equations may already be given in standard form.) It may be helpful to begin sketching the graph for part (h) as a visual aid to answer the questions below. (b) State the equation of the axis. (c) (d) (e) (f) (g) State the coordinates of the vertex. State the equation of the directrix. State the coordinates of the focus. State the focal width. State the coordinates of the endpoints of the focal chord. (h) Sketch a graph of the parabola which includes the features from (c)-(e) and (g). Label the vertex V and the focus F. 9. x2 4 y = 0 10. y 2 12 x = 0 11. 10x = y 2 26. y 2 + 10 y + x + 28 = 0 27. y 2 4 y + 2 x 4 = 0 28. 12 y + x 2 4 x = 16 29. 3 y 2 + 30 y 8 x + 67 = 0 30. 5 x 2 + 30 x 16 y = 19 31. 2 x 2 8 x + 7 y = 34 32. 4 y 2 8 y + 9 x + 40 = 0 Use the given features of each of the following parabolas to write an equation for the parabola in standard form. 33. Vertex: Focus: ( 2, 5 ) ( 4, 5) 34. Vertex: (1, 3) Focus: (1, 0 ) 12. 6 y = x 2 Math 1330, Precalculus The University of Houston Chapter 8: Analytic Geometry Exercise Set 8.1: Parabolas 35. Vertex: Focus: 36. Vertex: Focus: 37. Focus: ( 2, 0 ) 48. Endpoints of focal chord: ( 2, 4 ) ( 4, 2 ) ( 6, 2 ) Answer the following. 49. Write an equation of the line tangent to the parabola with equation f ( x ) = x 2 + 5 x + 4 at: ( 2, 3) (a) x = 3 (b) x = 2 ( 4, 1) Directrix: y = 5 39. Focus: 50. Write an equation of the line tangent to the parabola with equation f ( x ) = 3x 2 + 6 x + 1 at: ( 4, 1) (a) x = 0 (b) x = 1 Directrix: x = 7 12 40. Focus: ( 3, 5 ) 51. Write an equation of the line tangent to the parabola with equation f ( x ) = 2 x 2 + 5 x 1 at: Directrix: x = 4 41. Focus: (a) x = 1 1 (b) x = 2 ( 4, 2 ) Opens downward p=7 52. Write an equation of the line tangent to the parabola with equation f ( x ) = x 2 + 4 x 2 at: 42. Focus: (1, 5 ) Opens to the right p=3 43. Vertex: (a) x = 3 3 (b) x = 2 ( 5, 6 ) Opens upward Length of focal chord: 6 53. Write an equation of the line tangent to the parabola with equation f ( x ) = 4 x 2 5 x 3 at the point (1, 4 ) . 1 44. Vertex: 0, 2 Opens to the left Length of focal chord: 2 45. Vertex: 54. Write an equation of the line tangent to the parabola with equation f ( x ) = 2 x 2 + 5 x + 4 at the point ( 3, 7 ) . ( 3, 2 ) Horizontal axis Passes through ( 6, 5 ) 46. Vertex: 55. Write an equation of the line tangent to the parabola with equation f ( x ) = x 2 6 x + 1 at the point ( 5, 6 ) . ( 2, 1) Vertical axis Passes through ( 8, 5 ) 47. Endpoints of focal chord: and ( 6, 3) Opens downward Directrix: y = 9 38. Focus: ( 2, 3) 56. Write an equation of the line tangent to the parabola with equation f ( x ) = 3x 2 + 4 x + 9 at ( 0, 5) and ( 0, 5 ) the point ( 2, 5 ) . Opens to the left Math 1330, Precalculus The University of Houston Chapter 8: Analytic Geometry Exercise Set 8.1: Parabolas Give the point(s) of intersection of the parabola and the line whose equations are given. 57. f ( x ) = x 2 4 x + 11 g ( x ) = 5x 3 58. f ( x ) = x 2 8 x + 1 g ( x ) = 2 x 10 59. f ( x ) = x 2 + 10 x + 10 g ( x ) = 8x + 9 60. f ( x ) = x 2 5 x 10 g ( x ) = 7 x + 14 61. f ( x ) = x 2 + 6 x 5 g ( x ) = 6 x 14 62. f ( x ) = x 2 + 3x 2 g ( x ) = 5 x + 13 63. f ( x ) = 3 x 2 + 6 x + 1 g ( x ) = 3 x + 7 64. f ( x ) = 2 x 2 + 8 x 5 g ( x) = 6x 5 Math 1330, Precalculus The University of Houston Chapter 8: Analytic Geometry Exercise Set 8.2: Ellipses Write each of the following equations in the standard form for the equation of an ellipse, where the standard form is represented by one of the following equations: ( x h)2 ( y k )2 + a2 1. + b2 =1 ( x h)2 ( y k )2 + b2 + a2 (f) State the eccentricity. (g) Sketch a graph of the ellipse which includes the features from (b)-(e). Label the center C, and the foci F1 and F2. =1 11. x2 y2 + =1 9 49 12. x2 y2 + =1 36 4 25 x 2 + 4 y 2 100 = 0 2 2 2. 9 x + 16 y 144 = 0 3. 9 x 2 36 x + 4 y 2 32 y + 64 = 0 13. 4. 4 x 2 + 24 x + 16 y 2 32 y 12 = 0 5. 3x 2 + 2 y 2 30 x 12 y = 87 6. x 2 + 8 y 2 + 113 = 14 x + 48 y 7. 16 x 2 16 x 64 = 8 y 2 24 y + 42 8. 18 x 2 + 9 y 2 = 153 24 x + 6 y 15. Answer the following. (a) What is the equation for the eccentricity, e, of an ellipse? (b) As e approaches 1, the ellipse appears to become more (choose one): elongated circular (c) If e = 0 , the ellipse is a __________. 17. 18. 19. 10. The sum of the focal radii of an ellipse is always equal to __________. 20. Answer the following for each ellipse. For answers involving radicals, give exact answers and then round to the nearest tenth. (a) Write the given equation in the standard form for the equation of an ellipse. (Some equations may already be given in standard form.) It may be helpful to begin sketching the graph for part (g) as a visual aid to answer the questions below. 2 + 16 y2 =1 4 2 14. 16. 9. ( x 2) x 2 ( y + 1) + =1 9 5 ( x 2) 2 + 25 ( x + 5) 2 + 16 ( x + 4) 2 + 9 ( x + 2) 2 36 ( y + 3) 2 =1 16 ( y + 2) 2 =1 25 ( y 3) 2 =1 1 ( y + 3) + 2 =1 16 ( x 2 )2 ( y + 4 )2 + 11 ( x + 3) 20 2 + 36 ( y 5) =1 2 =1 4 21. 4 x 2 + 9 y 2 36 = 0 22. 4 x 2 + y 2 = 1 23. 25 x 2 + 16 y 2 311 = 50 x 64 y 24. 16 x 2 + 25 y 2 = 150 y + 175 25. 16 x 2 32 x + 4 y 2 40 y + 52 = 0 (b) State the coordinates of the center. (c) State the coordinates of the vertices, and then state the length of the major axis. (d) State the coordinates of the endpoints of the minor axis, and then state the length of the minor axis. (e) State the coordinates of the foci. Math 1330, Precalculus The University of Houston 26. 25 x 2 + 9 y 2 100 x + 54 y 44 = 0 27. 16 x 2 + 7 y 2 + 64 x 42 y + 15 = 0 28. 4 x 2 + 3 y 2 16 x + 6 y 29 = 0 Chapter 8: Analytic Geometry Exercise Set 8.2: Ellipses Use the given features of each of the the following ellipses to write an equation for the ellipse in standard form. 29. Center: ( 0, 0 ) a =8 b=5 Horizontal Major Axis 38. Foci: ( 0, 0 ) a=7 b=3 Vertical Major Axis 31. Center: ( 4, 7 ) a=5 b=3 Vertical Major Axis 32. Center: ( 2, 4 ) a =5 b=2 Horizontal Major Axis 39. Foci: ( 3, 5) Length of major axis = 6 Length of minor axis = 4 Horizontal Major Axis 34. Center: ( 2,1) and ( 7, 2 ) ( 3, 4 ) and ( 7, 4 ) Passes through the point ( 2,1) 41. Center: ( 5, 2 ) a =8 3 e= 4 Vertical major axis ( 4, 2 ) a=6 2 e= 3 Horizontal major axis 42. Center: e= ( 0, 4 ) and ( 0, 8 ) 1 3 44. Foci: (1, 5 ) and (1, 3) e= Length of major axis = 10 Length of minor axis = 2 Vertical Major Axis ( 1, 2 ) Passes through the point ( 3, 5) 43. Foci: 33. Center: and ( 2, 5 ) a =8 40. Foci: 30. Center: ( 2, 3) 1 2 45. Foci: ( 2, 3) and ( 6, 3) e = 0.4 35. Foci: ( 2, 5) and ( 2, 5 ) a =9 46. Foci: ( 2,1) and (10, 1) e = 0.8 36. Foci: ( 4, 3) and ( 4, 3) a=7 47. Foci: ( 3, 0 ) and ( 3, 0 ) Sum of the focal radii = 8 37. Foci: ( 8, 1) and ( 2, 1) a=6 48. Foci: ( 0, ) ( 11 and 0, 11 ) Sum of the focal radii = 12 Math 1330, Precalculus The University of Houston Chapter 8: Analytic Geometry Exercise Set 8.2: Ellipses A circle is a special case of an ellipse where a = b . It 2 2 2 2 60. Center: 2 then follows that c = a b = a a = 0 , so c = 0 . (Therefore, the foci are at the center of the circle, and this is simply labeled as the center and not the focus.) The standard form for the equation of a circle is ( x h) 2 2 + ( y k) = r 2 (Note: If each term in the above equation were divided by r 2 , it would look like the standard form for an ellipse, with a = b = r .) ( 3, 5) Circle is tangent to the y-axis Answer the following for each circle. For answers involving radicals, give exact answers and then round to the nearest tenth. (a) Write the given equation in the standard form for the equation of a circle. (Some equations may already be given in standard form.) (b) State the coordinates of the center. Using the above information, use the given features of each of the following circles to write an equation for the circle in standard form. 49. Center: ( 0, 0 ) Radius: 9 50. Center: Radius: 51. Center: ( 0, 0 ) (c) State the length of the radius. (d) Sketch a graph of the circle which includes the features from (b) and (c). Label the center C and show four points on the circle itself (these four points are equivalent to the vertices and the endpoints of the minor axis for an ellipse). 61. x 2 + y 2 36 = 0 5 62. x 2 + y 2 8 = 0 ( 7, 2 ) Radius: 10 63. ( 2, 5 ) Radius: 7 52. Center: 53. Center: ( 8, 0 ) ( x + 5 ) 2 + ( y 2 )2 = 4 66. ( x 1)2 + ( y 4 )2 = 36 67. ( x + 4 )2 + ( y + 3)2 = 12 68. ( x 1)2 + ( y 5 )2 = 7 69. x 2 + y 2 + 2 x 10 y + 17 = 0 ( 6, 3) 57. Endpoints of diameter: ( 4, 6 ) 58. Endpoints of diameter: ( 3, 0 ) ( 3, 5) Circle is tangent to the x-axis Math 1330, Precalculus The University of Houston =1 65. ( 2, 5 ) Passes through the point ( 8, 2 ) 59. Center: 16 2 x2 ( y + 2) + =1 9 9 ( 3, 4 ) Passes through the point ( 7, 6 ) 56. Center: ( y + 2) + 64. Radius: 2 5 55. Center: 16 2 2 Radius: 3 2 54. Center: ( x 3) 70. x 2 + y 2 + 6 x 2 y 6 = 0 and ( 2, 0 ) and ( 7,10 ) 71. x 2 + y 2 + 10 x 8 y + 36 = 0 72. x 2 + y 2 4 x 14 y + 50 = 0 73. 3 x 2 + 3 y 2 + 18 x 24 y + 63 = 0 74. 2 x 2 + 2 y 2 + 10 x + 12 y + 52 = 10 Chapter 8: Analytic Geometry Exercise Set 8.2: Ellipses Answer the following. 75. A circle passes through the points ( 7, 6 ) , ( 7, 2 ) and (1, 6 ) . Write the equation of the circle in standard form. 76. A circle passes through the points ( 4, 3) , ( 2, 3) and ( 2, 1) . Write the equation of the circle in standard form. 77. A circle passes through the points ( 2,1) , ( 2, 3) and ( 8, 1) . Write the equation of the circle in standard form. 78. A circle passes through the points ( 7, 8 ) , ( 7, 2 ) and ( 1, 2 ) . Write the equation of the circle in standard form. Math 1330, Precalculus The University of Houston Chapter 8: Analytic Geometry Math 1330 Precalculus Electronic Homework (EHW 10) Sections 8.1, 8.2 Work the following problems and choose the correct answer. The problems that refer to the Textbook may be found at www.casa.uh.edu in the Online Textbook under the Exercises link in the appropriate section. Record your answers by logging into your CASA account, go to the EMCF link in the top menu bar, selecting the appropriate assignment in the EMCF menu and recording your choices on the EMCF form. You MUST record your answers by the deadline given. No late homework will be accepted. Section 8.1 1. 2. Problem 8.1.20 e f A. (1, 6); 8 B. (1, 4); 4 C. (0, 3); 4 D. (1, 4); 8 E. None of the above x=2 B. (4, 0); y=3 (4, 2); y = -4 Problem 8.1.28 c d A. (2, -1); C. (5, 1); x=5 D. E. (2, -1); y=2 F. None of these 3. Problem 8.1.38 A. x2 C. ( x 4)2 E. None of the above 8( y 1) 8( y 3) B. ( x 1) 2 2( y 3) D. ( x 4) 2 2( y 3) Page 1 of 6 For numbers 4 - 5, match each equation to its graph. 4. y 2 A. 5. y 2 4 x 1 4 x 1 B. C. D. E. F. None of these Page 2 of 6 Which could be the graph of the parabola x 6. A. B. C. D. E. y2 2 y ? None of the above Page 3 of 6 Section 8.2 7. 8. Problem 8.2.38 A. ( x 2) 2 80 ( y 1) 2 64 1 B. ( x 2) 2 48 ( y 1) 2 64 1 C. ( x 2) 2 64 ( y 1) 2 16 1 D. ( x 2) 2 4 ( y 8) 2 25 1 E. None of the above Problem 8.2.54 A. x 8 2 y2 20 B. x 8 2 y2 2 5 C. x 8 2 y2 10 D. x 8 2 y2 20 E. 9. None of the above Problem 8.2.58 A. x 5 2 y 5 2 1 B. x 5 2 y 5 2 116 C. x 5 2 y 5 2 29 D. x 2 2 y 5 2 29 E. x 5 2 y 5 2 B. (2, 7); r =9 D. (2, 7); r=3 116 10. Problems 8.2.72 b c A. (-2, -7); r C. (-2, -7); r=3 E. (2, 7); r 3 3 F. None of these Page 4 of 6 For numbers 11 - 15, match the equation with the graph. The graphs are on the next page. Try drawing a quick sketch of each before looking at the answer choices. Each choice can be used more than once or not at all. 11. x 2 4 2 y 3 9 2 1 12. x 3 9 2 y 2 4 2 1 13. x 3 4 2 y 2 9 2 1 14. 4 x 2 16 x 9 y 2 54 y 61 0 15. 9 x 2 36 x 4 y 2 24 y 36 0 Page 5 of 6 A B. C D E F Page 6 of 6