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.  xy22  8  5 form for the equation of a parabola, where the standard form is represented by one of the following 2 14. yx4 16 equations:  

2 2 2  x h 4 p y  k   x h  4 p y  k 15. yx43  2 2  y k 4 p x  h  y k  4 p x  h 16. xy3412   

17. (xy 5)2   2  4 1. y2 14 y  2 x  43  0   2 2. x2 10 x  12 y   61 18.  yx1   10  3

2 3. 9y  x2  8 x  10 19.  yx6   1  2

4. 7x  y2  10 y  24 20.  xy12   8  6

2 5. x3 y  24 y  50 21. x2 12 x  6 y  24  0

2 6. y2 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. x25   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. xy40 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, 221 221 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. x8 y  113  14 x  48 y 95

22 7. 16x 16 x  64   8 y  24 y  42 xy2322  15. 1 25 16 8. 18x22 9 y  153  24 x  6 y xy5222  16. 1 Answer the following. 16 25 9. (a) What is the equation for the eccentricity, e, xy4322  of an ellipse? 17. 1 (b) As e approaches 1, the ellipse appears to 91 become more (choose one): 22 xy23   elongated circular 18. 1 (c) If e  0 , the ellipse is a ______. 36 16 22  xy24   10. The sum of the focal radii of an ellipse is always 19. 1 equal to ______. 11 36

xy3522  20. 1 Answer the following for each ellipse. For answers 20 4 involving radicals, give exact answers and then round 22 to the nearest tenth. 21. 4xy 9  36  0

(a) Write the given equation in the standard form 22 for the equation of an ellipse. (Some equations 22. 41xy may already be given in standard form.) 23. 25x22 16 y  311  50 x  64 y It may be helpful to begin sketching the graph for part (g) as a visual aid to answer the questions 22 24. 16x 25 y  150 y  175 below. (b) State the coordinates of the center. 25. 16x22 32 x  4 y  40 y  52  0 (c) State the coordinates of the vertices of the 22 major axis, and then state the length of the 26. 25x 9 y  100 x  54 y  44  0 major axis. 22 (d) State the coordinates of the vertices of the 27. 16x 7 y  64 x  42 y  15  0 minor axis, and then state the length of the minor axis. 28. 4x22 3 y  16 x  6 y  29  0

MATH 1330 Precalculus 673 Exercise Set 8.2: Ellipses

Use the given features of each of the the following 38. Foci: 2, 3 and 2, 5 ellipses to write an equation for the ellipse in standard a  8 form.

29. Center: 0, 0 39. Foci: 1, 2 and 7, 2 a  8 Passes through the point 3, 5 b  5 Horizontal Major Axis 40. Foci: 3, 4 and 7, 4 30. Center: Passes through the point 2,1 a  7 b  3 41. Center: 5, 2 Vertical Major Axis a  8 3 e  31. Center: 4, 7 4 a  5 Vertical major axis b  3 Vertical Major Axis 42. Center: 4, 2   a  6 32. Center: 2, 4 2 e  a  5 3 b  2 Horizontal major axis Horizontal Major Axis

43. Foci: 0, 4 and 0, 8 33. Center: 3, 5 1 e  Length of major axis  6 3 Length of minor axis  4 Horizontal Major Axis 44. Foci: 1, 5 and 1, 3 1 34. Center: 2,1 e  2 Length of major axis 10 Length of minor axis  2 Vertical Major Axis 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

674 University of Houston Department of Mathematics Exercise Set 8.2: Ellipses

A circle is a special case of an ellipse where ab . It Circle is tangent to the y-axis then follows that c2 a 2  b 2  a 2  a 2  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.) Answer the following for each circle. For answers involving radicals, give exact answers and then round The standard form for the equation of a circle is to the nearest tenth. 222  x h  y  k  r (a) Write the given equation in the standard form for the equation of a circle. (Some equations (Note: If each term in the above equation were may already be given in standard form.) divided by r 2 , it would look like the standard form for an ellipse, with a b r .) (b) State the coordinates of the center. (c) State the length of the radius. Using the above information, use the given features of (d) Sketch a graph of the circle which includes each of the following circles to write an equation for the features from (b) and (c). Label the center the circle in standard form. C and show four points on the circle itself (these four points are equivalent to the 49. Center: 0, 0 vertices of the major and minor axes for an ellipse). Radius: 9

61. xy22 36  0 50. Center: 0, 0 Radius: 5 62. xy22 80 

51. Center: 7, 2 xy3222  63. 1 Radius: 10 16 16

2 52. Center: 2, 5 x2  y  2 64. 1 Radius: 7 99

22 53. Center: 3, 4 65. xy5   2  4

Radius: 32 22 66. xy1   4  36 54. Center: 8, 0 67. xy422   3  12 Radius: 25

68. xy122   5  7 55. Center: 2, 5    

Passes through the point 7, 6 69. x22 y 2 x  10 y  17  0

56. Center: 6, 3 70. x22 y 6 x  2 y  6  0 Passes through the point 8, 2 22 71. x y 10 x  8 y  36  0 57. Endpoints of diameter: 4, 6 and 2, 0 72. x22 y 4 x  14 y  50  0

58. Endpoints of diameter: 3, 0 and 7,10 22     73. 3x 3 y  18 x  24 y  63  0

59. Center: 3, 5 74. 2x22 2 y  10 x  12 y  52  10 Circle is tangent to the x-axis

60. Center: MATH 1330 Precalculus 675 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.

676 University of Houston Department of Mathematics SECTION 8.3 Hyperbolas

Section 8.3: Hyperbolas

 Equations of Hyperbolas with Center at the Origin

Equations of Hyperbolas with Center at the Origin

Definition of a Hyperbola:

Equations of Hyperbolas with Center at the Origin:

MATH 1330 Precalculus 677 CHAPTER 8 Analytic Geometry

678 University of Houston Department of Mathematics SECTION 8.3 Hyperbolas

MATH 1330 Precalculus 679 CHAPTER 8 Analytic Geometry

Example:

Solution:

680 University of Houston Department of Mathematics SECTION 8.3 Hyperbolas

Example:

MATH 1330 Precalculus 681 CHAPTER 8 Analytic Geometry

Solution:

682 University of Houston Department of Mathematics SECTION 8.3 Hyperbolas

The Standard Form for the Equation of a Hyperbola:

MATH 1330 Precalculus 683 CHAPTER 8 Analytic Geometry

Example:

Solution:

684 University of Houston Department of Mathematics SECTION 8.3 Hyperbolas

MATH 1330 Precalculus 685 CHAPTER 8 Analytic Geometry

Example:

Solution:

686 University of Houston Department of Mathematics SECTION 8.3 Hyperbolas

Example:

Solution: Part (a):

Part (b):

Part (c):

MATH 1330 Precalculus 687 CHAPTER 8 Analytic Geometry

Part (d):

Additional Example 1:

Solution:

688 University of Houston Department of Mathematics SECTION 8.3 Hyperbolas

Additional Example 2:

Solution:

MATH 1330 Precalculus 689 CHAPTER 8 Analytic Geometry

Additional Example 3:

Solution:

690 University of Houston Department of Mathematics SECTION 8.3 Hyperbolas

Additional Example 4:

Solution:

MATH 1330 Precalculus 691 CHAPTER 8 Analytic Geometry

692 University of Houston Department of Mathematics Exercise Set 8.3: Hyperbolas

Identify the type of conic section (parabola, ellipse, 17. The following questions establish the formulas circle, or hyperbola) represented by each of the for the slant asymptotes of following equations. (In the case of a circle, identify 22 the conic section as a circle rather than an ellipse.) Do  y k  x h 1 . NOT write the equations in standard form; these ab22 questions can instead be answered by looking at the signs of the quadratic terms. (a) State the point-slope equation for a line. (b) Substitute the center of the hyperbola, 1. 2y x2  9 x  0 hk,  into the equation from part (a). 2. 14x22 7 x  12 y   6 y  95 (c) Recall that the formula for slope is 22 rise 3. 7x 3 y  5 x  y  40 represented by . In the equation run 4. y2 99  y  x , what is the “rise” of 5. 3x22 7 x  3 y   12 y  13 each slant asymptote from the center? What 22 is the “run” of each slant asymptote from the 6. x10 x   2 y  y  5 center? 7. 4y22 2 x  8 y  6 x  9 (d) Based on the answers to part (c), what is the slope of each of the asymptotes for the graph 8. 8y22 24 x  8 x  30 of ? (Remember that

there are two slant asymptotes passing Write each of the following equations in the standard through the center of the hyperbola, one form for the equation of a hyperbola, where the having positive slope and one having standard form is represented by one of the following negative slope.) equations:

22 22 (e) Substitute the slopes from part (d) into the  x h  y k   y k  x h equation from part (b) to obtain the 1 1 ab22 ab22 equations of the slant asymptotes.

9. yx228  8  0 18. The following questions establish the formulas for the slant asymptotes of 10. 3xy22 10  30  0 x h22 y k  22 1 . 11. x y 6 x   2 y  3 ab22 12. 9x22 3 y  48 y  192 (a) State the point-slope equation for a line. (b) Substitute the center of the hyperbola, 13. 7x22 5 y  14 x  20 y  48  0 into the equation from part (a). 22 14. 9y 2 x  90 y  16 x  175  0 (c) Recall that the formula for slope is represented by . In the equation Answer the following. 15. The length of the transverse axis of a hyperbola , what is the “rise” of is ______. each slant asymptote from the center? What 16. The length of the conjugate axis of a hyperbola is 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 693 Exercise Set 8.3: Hyperbolas

x h22 y k  xy22 of 1 ? (Remember that 22. 1 ab22 36 25 there are two slant asymptotes passing 23. 9xy22 25  225  0 through the center of the hyperbola, one having positive slope and one having 24. 16yx22  16  0 negative slope.) (e) Substitute the slopes from part (d) into the  xy1522  25. 1 equation from part (b) to obtain the 16 9 equations of the slant asymptotes. xy5222  26. 1 19. In the standard form for the equation of a 4 16 hyperbola, a2 represents (choose one):  yx3122  the larger denominator 27. 1 the denominator of the first term 25 36 22  yx64   20. In the standard form for the equation of a 28. 1 64 36 hyperbola, b2 represents (choose one): 22 the smaller denominator 29. x25 y  8 x  150 y  234  0

the denominator of the second term 22 30. 4y 81 x   162 x  405 Answer the following for each hyperbola. For answers 31. 64x22 9 y  18 y  521  128 x involving radicals, give exact answers and then round to the nearest tenth. 32. 16x22 9 y  64 x  18 y  89  0 (a) Write the given equation in the standard form 33. 5y22 4 x  50 y  24 x  69  0 for the equation of a hyperbola. (Some equations may already be given in standard 22 34. 7x 9 y  72 y  32  70 x form.) It may be helpful to begin sketching the graph for 35. x223 y  18 x  27 part (h) as a visual aid to answer the questions below. 36. 4y22 21 x  8 y  42 x  89  0 (b) State the coordinates of the center. (c) State the coordinates of the vertices, and then Use the given features of each of the following state the length of the transverse axis. hyperbolas to write an equation for the hyperbola in standard form. (d) State the coordinates of the endpoints of the conjugate axis, and then state the length of the 37. Center: 0, 0 conjugate axis. a  8 (e) State the coordinates of the foci. b  5 (f) State the equations of the asymptotes. Horizontal Transverse Axis (Answers may be left in point-slope form.) (g) State the eccentricity. 38. Center: (h) Sketch a graph of the hyperbola which a  7 includes the features from (b)-(f), along with b  3 the central rectangle. Label the center C, the Vertical Transverse Axis vertices V1 and V2, and the foci F1 and F2. 39. Center: 2, 5 yx22 21. 1 a  2 9 49 b 10 Vertical Transverse Axis

694 University of Houston Department of Mathematics Exercise Set 8.3: Hyperbolas

40. Center: 3, 4 52. Center: a 1 Vertex: b  6 Equation of one asymptote: Horizontal Transverse Axis 6xy 5   7

41. Center: 6,1 53. Vertices: 1, 4 and 7, 4 Length of transverse axis: 10     Length of conjugate axis: 8 e  3 Vertical Transverse Axis 54. Vertices: 2, 6 and 2, 1 42. Center: 2, 5 7 e  Length of transverse axis: 6 3 Length of conjugate axis: 14 Horizontal Transverse Axis 55. Center: 4, 3

One focus is at 4, 6 43. Foci: 0, 9 and 0, 9   3 Length of transverse axis: 6 e  2 44. Foci: 5, 0 and 5, 0 56. Center: 1, 2 Length of conjugate axis: 4   One focus is at 9, 2 45. Foci: 2, 3 and 10, 3 5 e  Length of conjugate axis: 10 2

46. Foci: 3, 8 and 3, 6 57. Foci: 3, 0 and 3, 8 Length of transverse axis: 8 4 e  3 47. Vertices: 4, 7 and 4, 9 b  4 58. Foci: 6, 5 and 6, 7 e  2 48. Vertices: 1, 6 and 7, 6 b  7

49. Center: 5, 3 One focus is at 5, 8 One vertex is at 5, 6

50. Center: 3, 4 One focus is at 7, 4 One vertex is at 5, 4

51. Center: 1, 2 Vertex: 3, 2 Equation of one asymptote: 7xy 4   15

MATH 1330 Precalculus 695