Exercise Set 8.3: Hyperbolas Identify the type of conic section (parabola, ellipse, circle, or hyperbola) represented by each of the following equations. (In the case of a circle, identify the conic section as a circle rather than an ellipse.) Do NOT write the equations in standard form; these questions can instead be answered by looking at the signs of the quadratic terms. 2 y + x2 + 9 x = 0 2. 14 x 2 + 7 x 12 y = 6 y 2 + 95 a2 3. 7 x 3 y = 5 x y + 40 4. y2 + 9 = 9 y x (a) State the point-slope equation for a line. = 1 , what is the \"rise\" of a2 b2 each slant asymptote from the center? What is the \"run\" of each slant asymptote from the center? 2 3 x 7 x + 3 y = 12 y + 13 6. x 2 + 10 x = 2 y y 2 + 5 7. 4 y 2 + 2x2 = 8 y 6 x + 9 8. 8 y 2 + 24 x = 8 x 2 + 30 (d) Based on the answers to part (c), what is the slope of each of the asymptotes for the graph Write each of the following equations in the standard form for the equation of a hyperbola, where the standard form is represented by one of the following equations: a2 9. ( y k) b2 2 =1 ( y k) 2 a2 ( x h) b2 2 =1 y 2 8x2 8 = 0 10. 3x 2 10 y 2 30 = 0 2 2 11. x y 6 x = 2 y 3 ( y k )2 ( x h )2 = 1 ? (Remember that a2 b2 there are two slant asymptotes passing through the center of the hyperbola, one having positive slope and one having negative slope.) of 2 =1 . ( y k )2 ( x h )2 5. ( x h) b2 (c) Recall that the formula for slope is rise represented by . In the equation run 2 2 ( y k )2 ( x h )2 (b) Substitute the center of the hyperbola, ( h, k ) into the equation from part (a). 1. 2 17. The following questions establish the formulas for the slant asymptotes of (e) Substitute the slopes from part (d) into the equation from part (b) to obtain the equations of the slant asymptotes. 18. The following questions establish the formulas for the slant asymptotes of ( x h )2 ( y k )2 a2 b2 =1 . 12. 9 x 2 3 y 2 = 48 y + 192 (a) State the point-slope equation for a line. 13. 7 x 2 5 y 2 + 14 x + 20 y 48 = 0 (b) Substitute the center of the hyperbola, ( h, k ) into the equation from part (a). 14. 9 y 2 2 x 2 + 90 y + 16 x + 175 = 0 Answer the following. 15. The length of the transverse axis of a hyperbola is __________. 16. The length of the conjugate axis of a hyperbola is __________. (c) Recall that the formula for slope is rise represented by . In the equation run ( x h )2 ( y k )2 = 1 , what is the \"rise\" of a2 b2 each slant asymptote from the center? What is the \"run\" of each slant asymptote from the center? (d) Based on the answers to part (c), what is the slope of each of the asymptotes for the graph Math 1330, Precalculus The University of Houston Chapter 8: Analytic Geometry Exercise Set 8.3: Hyperbolas ( x h )2 ( y k )2 = 1 ? (Remember that a2 b2 there are two slant asymptotes passing through the center of the hyperbola, one having positive slope and one having negative slope.) of (e) Substitute the slopes from part (d) into the equation from part (b) to obtain the equations of the slant asymptotes. 19. In the standard form for the equation of a hyperbola, a 2 represents (choose one): the larger denominator the denominator of the first term 20. In the standard form for the equation of a hyperbola, b 2 represents (choose one): the smaller denominator the denominator of the second term 22. x2 y 2 =1 36 25 23. 9 x 2 25 y 2 225 = 0 24. 16 y 2 x 2 16 = 0 25. ( x + 1)2 ( y 5 )2 16 =1 9 26. ( x + 5 )2 ( y + 2 )2 =1 27. ( y 3)2 ( x 1)2 =1 28. ( y 6 )2 ( x + 4 )2 =1 4 25 64 16 36 36 29. x 2 25 y 2 + 8 x 150 y 234 = 0 30. 4 y 2 81x 2 = 162 x + 405 Answer the following for each hyperbola. For answers involving radicals, give exact answers and then round to the nearest tenth. 31. 64 x 2 9 y 2 + 18 y = 521 128 x (a) Write the given equation in the standard form for the equation of a hyperbola. (Some equations may already be given in standard form.) 33. 5 y 2 4 x 2 50 y 24 x + 69 = 0 It may be helpful to begin sketching the graph for part (h) as a visual aid to answer the questions below. 35. x 2 3 y 2 = 18 x + 27 32. 16 x 2 9 y 2 64 x 18 y 89 = 0 34. 7 x 2 9 y 2 72 y = 32 70 x 36. 4 y 2 21x 2 8 y 42 x 89 = 0 (b) State the coordinates of the center. (c) State the coordinates of the vertices, and then state the length of the transverse axis. (d) State the coordinates of the endpoints of the conjugate axis, and then state the length of the conjugate axis. (e) State the coordinates of the foci. (f) State the equations of the asymptotes. (Answers may be left in point-slope form.) (g) State the eccentricity. (h) Sketch a graph of the hyperbola which includes the features from (b)-(f), along with the central rectangle. Label the center C, the vertices V1 and V2, and the foci F1 and F2. y 2 x2 =1 21. 9 49 Math 1330, Precalculus The University of Houston Use the given features of each of the following hyperbolas to write an equation for the hyperbola in standard form. 37. Center: ( 0, 0 ) a =8 b=5 Horizontal Transverse Axis 38. Center: ( 0, 0 ) a=7 b=3 Vertical Transverse Axis 39. Center: ( 2, 5) a=2 b = 10 Vertical Transverse Axis Chapter 8: Analytic Geometry Exercise Set 8.3: Hyperbolas 40. Center: ( 3, 4 ) a =1 b=6 Horizontal Transverse Axis 41. Center: ( 6, 1) Length of transverse axis: 10 Length of conjugate axis: 8 Vertical Transverse Axis 42. Center: ( 0, 9 ) and ( 0, 9 ) Length of transverse axis: 6 44. Foci: ( 5, 0 ) and ( 5, 0 ) Length of conjugate axis: 4 45. Foci: ( 2, 3) and (10, 3) Length of conjugate axis: 10 46. Foci: ( 3, 8 ) and ( 3, 6 ) Length of transverse axis: 8 47. Vertices: Vertex: ( 1, 2 ) ( 3, 2 ) Equation of one asymptote: 6 x + 5 y = 7 53. Vertices: ( 4, 7 ) and ( 4, 9 ) b=4 e= and ( 7, 4 ) ( 2, 6 ) and ( 2, 1) 7 3 ( 4, 3) One focus is at ( 4, 6 ) 55. Center: e= 3 2 ( 1, 2 ) One focus is at ( 9, 2 ) 56. Center: e= 5 2 57. Foci: e= ( 3, 0 ) and ( 3, 8 ) 4 3 58. Foci: 48. Vertices: (1, 6 ) and ( 7, 6 ) ( 1, 4 ) e=3 54. Vertices: ( 2, 5) Length of transverse axis: 6 Length of conjugate axis: 14 Horizontal Transverse Axis 43. Foci: 52. Center: ( 6, 5) and ( 6, 7 ) e=2 b=7 49. Center: ( 5, 3) One focus is at ( 5, 8 ) One vertex is at ( 5, 6 ) ( 3, 4 ) One focus is at ( 7, 4 ) One vertex is at ( 5, 4 ) 50. Center: 51. Center: Vertex: ( 1, 2 ) ( 3, 2 ) Equation of one asymptote: 7 x 4 y = 15 Math 1330, Precalculus The University of Houston Chapter 8: Analytic Geometry Math 1330 Precalculus Electronic Homework (EHW 11) Section 8.3, Systems 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 8.3 1. Problem 8.3.22 parts (e) and (f) 5 x 6 A. e: (0, 36) and (0, 25) ; f: y B. e: (0, C. e: ( 61, 0) and ( 61, 0) ; f: y 6 x 5 D. e: ( 61, 0) and ( 61, 0) ; f: y 5 x 6 E. e: (6, 0) and (5, 0); 5 x 6 61) and (0, 61) ; f: y f: y 6 x 5 F: None of the above 2. Problem 8.3.24 (b) and (f) A. (4, 0) y 1 x 4 B. (0, 0) y 1 x 4 C. (0, 0) 4x D. (4, 0) 4x E. None of the above y y Page 1 of 4 3. Problem 8.3.30 c A. (-1, 0), (3, 0); 4 B. (0, 10), (0, -8); 18 C. (1, 9), (1, -9); 18 D. (9, 1), (-9, 1); 18 E. None of the above Systems 4. Solve the system of equations: 4x 2 9y2 4x 3y 2 A. (-3, 2) and (-3, -2) B. (3, 2), (3,-2), (-6, 2 2 ) and (-6, -2 2 ) C. (3, 2), (3, -2), (6, 2 2 ) and (6, -2 2 ) D. (3, 2) and (3, -2) 72 0 E. No solution F. None of these 5. In this problem, determine the number of points of intersection for the system. x 2 y 2 25 x y 8 A. 0 B. 1 C. 2 D. 3 E. 4 F. None of these 6. Graph and solve the system of equations: x y2 4 y x y 0 A. (-3, 3) and (3, -3) B. (3, -3), (-3,3), (0, 0) Page 2 of 4 C. (-3, 3) and (0,0) D. (3, -3), (-3,3), (0, 0), (-3,1) E. No solution F. None of these 7. Graph and solve the system of equations: x y2 6 y x 2y 5 A. (7, -1), (-5,5), (-7,1) B. (7, -1) and (5,5) C. (7, -1), (5,5) D. (-7, -1), (5,5) E. No solution F. None of these Problems 8-15: For each equation, determine which description listed below best describes the graph of the equation. Note: the answer choices given may be used more than once or not at all. A. B. C. D. E. F. G. H. Parabola Point Circle Hyperbola Ellipse Two intersecting lines One line No graph Page 3 of 4 8. 3x 2 12 y 2 9 x 12 y 9 10. 4x 2 0 12. 14. 9y2 x 2 5 x2 2 5y2 y 5 5 8 0 9. x 1 11. 5x2 13. x 2 y 3 2 y2 2 0 27 x 16 y 19 0 2 1 15. y2 3y 2 4y 7 2 12 x 1 2 72 Page 4 of 4