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# 38 ellipses and hyperbolas

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### 38 ellipses and hyperbolas

1. 1. Conic Sections
2. 2. Conic Sections Conic sections are the cross sections of right circular cones.
3. 3. Conic Sections Conic sections are the cross sections of right circular cones. A right circular cone
4. 4. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: A right circular cone
5. 5. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles A right circular cone
6. 6. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles • ellipses A right circular cone
7. 7. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles • ellipses • parabolas A right circular cone
8. 8. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles • ellipses • parabolas • hyperbolas A right circular cone
9. 9. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles • ellipses • parabolas • hyperbolas Where as straight lines are the graphs of 1st degree equations Ax + By = C, conic sections are the graphs of 2nd degree equations in x and y. A right circular cone
10. 10. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles • ellipses • parabolas • hyperbolas Where as straight lines are the graphs of 1st degree equations Ax + By = C, conic sections are the graphs of 2nd degree equations in x and y. In particular, the conic sections that are parallel to the axes (not tilted) have equations of the form Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers. A right circular cone
11. 11. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles • ellipses • parabolas • hyperbolas Where as straight lines are the graphs of 1st degree equations Ax + By = C, conic sections are the graphs of 2nd degree equations in x and y. In particular, the conic sections that are parallel to the axes (not tilted) have equations of the form Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers. We are to match these 2nd degree equations with the different conic sections. A right circular cone
12. 12. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles • ellipses • parabolas • hyperbolas Where as straight lines are the graphs of 1st degree equations Ax + By = C, conic sections are the graphs of 2nd degree equations in x and y. In particular, the conic sections that are parallel to the axes (not tilted) have equations of the form Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers. We are to match these 2nd degree equations with the different conic sections. The algebraic technique that enable us to sort out which equation corresponds to which conic section is called "completing the square". A right circular cone
13. 13. Conic Sections Conic sections are the cross sections of right circular cones. There are four different types of curves: • circles • ellipses • parabolas • hyperbolas Where as straight lines are the graphs of 1st degree equations Ax + By = C, conic sections are the graphs of 2nd degree equations in x and y. In particular, the conic sections that are parallel to the axes (not tilted) have equations of the form Ax2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers. We are to match these 2nd degree equations with the different conic sections. The algebraic technique that enable us to sort out which equation corresponds to which conic section is called "completing the square". We start with the Distance Formula. A right circular cone
14. 14. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: Conic Sections
15. 15. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 Conic Sections
16. 16. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 Conic Sections
17. 17. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 where Conic Sections Δy = the difference between the y's = y2 – y1
18. 18. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 where Conic Sections Δy = the difference between the y's = y2 – y1 Δx = the difference between the x's = x2 – x1
19. 19. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 where Example A. Find the distance between (2, –1) and (–2, 2). Conic Sections Δy = the difference between the y's = y2 – y1 Δx = the difference between the x's = x2 – x1
20. 20. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 where Example A. Find the distance between (2, –1) and (–2, 2). Δy = (–1) – (2) = –3 Δy=-3 Conic Sections Δy = the difference between the y's = y2 – y1 Δx = the difference between the x's = x2 – x1
21. 21. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 where Example A. Find the distance between (2, –1) and (–2, 2). Δy = (–1) – (2) = –3 Δx = (2) – (–2) = 4 Δy=-3 Δx=4 Conic Sections Δy = the difference between the y's = y2 – y1 Δx = the difference between the x's = x2 – x1
22. 22. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 where Example A. Find the distance between (2, –1) and (–2, 2). Δy = (–1) – (2) = –3 Δx = (2) – (–2) = 4 r = (–3)2 + 42 = 25 = 5 Δy=-3 Δx=4 r=5 Conic Sections Δy = the difference between the y's = y2 – y1 Δx = the difference between the x's = x2 – x1
23. 23. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 where Example A. Find the distance between (2, –1) and (–2, 2). Δy = (–1) – (2) = –3 Δx = (2) – (–2) = 4 r = (–3)2 + 42 = 25 = 5 Δy=-3 Δx=4 r=5 Conic Sections The geometric definition of all four types of conic sections are distance relations between points. Δy = the difference between the y's = y2 – y1 Δx = the difference between the x's = x2 – x1
24. 24. The Distance Formula: Given two points P = (x1, y1) and Q = (x2, y2) in the xy-plane, the distance r between P and Q is: r = (y2 – y1)2 + (x2 – x1)2 = Δy2 + Δx2 where Example A. Find the distance between (2, –1) and (–2, 2). Δy = (–1) – (2) = –3 Δx = (2) – (–2) = 4 r = (–3)2 + 42 = 25 = 5 Δy=-3 Δx=4 r=5 Conic Sections The geometric definition of all four types of conic sections are distance relations between points. We start with the circles. Δy = the difference between the y's = y2 – y1 Δx = the difference between the x's = x2 – x1
25. 25. Circles A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
26. 26. rr Circles A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center. C
27. 27. rr The radius and the center completely determine the circle. Circles center A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
28. 28. r The radius and the center completely determine the circle. Circles Let (h, k) be the center of a circle and r be the radius. (h, k) A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
29. 29. r The radius and the center completely determine the circle. Circles (x, y) Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on the circle, then the distance between (x, y) and the center is r. (h, k) A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
30. 30. r The radius and the center completely determine the circle. Circles (x, y) Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on the circle, then the distance between (x, y) and the center is r. Hence, (h, k) r = (x – h)2 + (y – k)2 A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
31. 31. r The radius and the center completely determine the circle. Circles (x, y) Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on the circle, then the distance between (x, y) and the center is r. Hence, (h, k) r = (x – h)2 + (y – k)2 or r2 = (x – h)2 + (y – k)2 A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
32. 32. r The radius and the center completely determine the circle. Circles (x, y) Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on the circle, then the distance between (x, y) and the center is r. Hence, (h, k) r = (x – h)2 + (y – k)2 or r2 = (x – h)2 + (y – k)2 This is called the standard form of circles. A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
33. 33. r The radius and the center completely determine the circle. Circles (x, y) Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on the circle, then the distance between (x, y) and the center is r. Hence, (h, k) r = (x – h)2 + (y – k)2 or r2 = (x – h)2 + (y – k)2 This is called the standard form of circles. Given an equation of this form, we can easily identify the center and the radius. A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
34. 34. r2 = (x – h)2 + (y – k)2 Circles
35. 35. r2 = (x – h)2 + (y – k)2 must be “ – ” Circles
36. 36. r2 = (x – h)2 + (y – k)2 r is the radius must be “ – ” Circles
37. 37. r2 = (x – h)2 + (y – k)2 r is the radius must be “ – ” (h, k) is the center Circles
38. 38. r2 = (x – h)2 + (y – k)2 r is the radius must be “ – ” (h, k) is the center Circles Example B. Write the equation of the circle as shown.
39. 39. r2 = (x – h)2 + (y – k)2 r is the radius must be “ – ” (h, k) is the center Circles Example B. Write the equation of the circle as shown. The center is (–1, 3) and the radius is 5. (–1, 3)
40. 40. r2 = (x – h)2 + (y – k)2 r is the radius must be “ – ” (h, k) is the center Circles Example B. Write the equation of the circle as shown. The center is (–1, 3) and the radius is 5. Hence the equation is: 52 = (x – (–1))2 + (y – 3)2 (–1, 3)
41. 41. r2 = (x – h)2 + (y – k)2 r is the radius must be “ – ” (h, k) is the center Circles Example B. Write the equation of the circle as shown. The center is (–1, 3) and the radius is 5. Hence the equation is: 52 = (x – (–1))2 + (y – 3)2 or 25 = (x + 1)2 + (y – 3 )2 (–1, 3)
42. 42. Example C. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it. Circles
43. 43. Example C. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the standard form: 42 = (x – 3)2 + (y – (–2))2 Circles
44. 44. Example C. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the standard form: 42 = (x – 3)2 + (y – (–2))2 Hence r = 4, center = (3, –2) Circles
45. 45. Example C. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the standard form: 42 = (x – 3)2 + (y – (–2))2 Hence r = 4, center = (3, –2) (3,-2) Circles
46. 46. Example C. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the standard form: 42 = (x – 3)2 + (y – (–2))2 Hence r = 4, center = (3, –2) (3,-2) Circles When equations are not in the standard form, we have to rearrange them into the standard form. We do this by "completing the square".
47. 47. Example C. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the standard form: 42 = (x – 3)2 + (y – (–2))2 Hence r = 4, center = (3, –2) (3,-2) Circles When equations are not in the standard form, we have to rearrange them into the standard form. We do this by "completing the square". To complete the square means to add a number to an expression so the sum is a perfect square.
48. 48. Example C. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the standard form: 42 = (x – 3)2 + (y – (–2))2 Hence r = 4, center = (3, –2) (3,-2) Circles When equations are not in the standard form, we have to rearrange them into the standard form. We do this by "completing the square". To complete the square means to add a number to an expression so the sum is a perfect square. This procedure is the main technique in dealing with 2nd degree equations.
49. 49. (Completeing the Square) Circles
50. 50. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, Circles
51. 51. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles
52. 52. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2
53. 53. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2
54. 54. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
55. 55. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2 b. y2 + 12y + (12/2)2
56. 56. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2 b. y2 + 12y + (12/2)2
57. 57. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2 b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
58. 58. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2 b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2 The following are the steps in putting a 2nd degree equation into the standard form.
59. 59. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2 b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2 The following are the steps in putting a 2nd degree equation into the standard form. 1. Group the x2 and the x-terms together, group the y2 and y terms together, and move the number term the the other side of the equation.
60. 60. (Completeing the Square) If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2. Circles Example D. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2 b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2 The following are the steps in putting a 2nd degree equation into the standard form. 1. Group the x2 and the x-terms together, group the y2 and y terms together, and move the number term the the other side of the equation. 2. Complete the square for the x-terms and for the y-terms. Make sure add the necessary numbers to both sides.
61. 61. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. Circles
62. 62. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: Circles
63. 63. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 Circles
64. 64. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 ;complete squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 Circles
65. 65. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 ;complete squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 Circles
66. 66. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 ;complete squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 Circles
67. 67. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 ;complete squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Circles
68. 68. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 ;complete squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Hence the center is (3 , –6), and radius is 3. Circles
69. 69. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 ;complete squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Hence the center is (3 , –6), and radius is 3. Circles
70. 70. Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left anf right most points. Graph it. We use completeing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 ;complete squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Hence the center is (3 , –6), and radius is 3. Circles
71. 71. Ellipses
72. 72. Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
73. 73. F2F1 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
74. 74. F2F1 P Q R If P, Q, and R are any points on a ellipse, Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
75. 75. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
76. 76. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 q1 q2 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
77. 77. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 q1 q2 r2r1 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
78. 78. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 = a constant q1 q2 r2r1 Ellipses Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
79. 79. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 = a constant q1 q2 r2r1 Ellipses An ellipse has a center (h, k ); (h, k) (h, k) Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
80. 80. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 = a constant q1 q2 r2r1 Ellipses An ellipse has a center (h, k ); it has two axes, the major (long) (h, k) (h, k) Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. Major axis Major axis
81. 81. F2F1 P Q R p1 p2 If P, Q, and R are any points on a ellipse, then p1 + p2 = q1 + q2 = r1 + r2 = a constant q1 q2 r2r1 Ellipses An ellipse has a center (h, k ); it has two axes, the major (long) and the minor (short) axes. (h, k)Major axis Minor axis (h, k) Major axis Minor axis Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
82. 82. These axes correspond to the important radii of the ellipse. Ellipses
83. 83. These axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius Ellipses x-radius x-radius
85. 85. These axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius. Ellipses x-radius The general equation for ellipses is Ax2 + By2 + Cx + Dy = E where A and B are the same sign but different numbers. x-radius y-radiusy-radius
86. 86. These axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius. Ellipses x-radius The general equation for ellipses is Ax2 + By2 + Cx + Dy = E where A and B are the same sign but different numbers. Using completing the square, such equations may be transform to the standard form of ellipses below. x-radius y-radiusy-radius
87. 87. (x – h)2 (y – k)2 a2 b2 Ellipses + = 1 The Standard Form (of Ellipses)
88. 88. (x – h)2 (y – k)2 a2 b2 Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
89. 89. (x – h)2 (y – k)2 a2 b2 (h, k) is the center. Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
90. 90. (x – h)2 (y – k)2 a2 b2 x-radius = a (h, k) is the center. Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
91. 91. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. The Standard Form (of Ellipses)
92. 92. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The Standard Form (of Ellipses)
93. 93. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The Standard Form (of Ellipses)
94. 94. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The Standard Form (of Ellipses)
95. 95. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. The Standard Form (of Ellipses)
96. 96. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. The Standard Form (of Ellipses)
97. 97. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) (7, -1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. So the right point is (7, –1), The Standard Form (of Ellipses)
98. 98. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) (7, -1) (3, 1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. So the right point is (7, –1), the top point is (3, 1), The Standard Form (of Ellipses)
99. 99. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) (7, -1)(-1, -1) (3, -3) (3, 1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. So the right point is (7, –1), the top point is (3, 1), the left and bottom points are (1, –1) and (3, –3). The Standard Form (of Ellipses)
100. 100. (x – h)2 (y – k)2 a2 b2 x-radius = a y-radius = b (h, k) is the center. Ellipses + = 1 This has to be 1. (3, -1) (7, -1)(-1, -1) (3, -3) (3, 1) Example A. Find the center, major and minor axes. Draw and label the top, bottom, right and left most points. (x – 3)2 (y + 1)2 42 22+ = 1 The center is (3, –1). The x-radius is 4. The y-radius is 2. So the right point is (7, –1), the top point is (3, 1), the left and bottom points are (–1, –1) and (3, –3). The Standard Form (of Ellipses)
101. 101. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Ellipses
102. 102. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: Ellipses
103. 103. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 Ellipses
104. 104. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 Ellipses
105. 105. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square Ellipses
106. 106. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 Ellipses
107. 107. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 +9 Ellipses
108. 108. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 +9 +16 Ellipses
109. 109. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 Ellipses
110. 110. Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36
111. 111. 9(x – 1)2 4(y – 2)2 36 36 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1
112. 112. 9(x – 1)2 4(y – 2)2 36 4 36 9 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1
113. 113. 9(x – 1)2 4(y – 2)2 36 4 36 9 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 (x – 1)2 (y – 2)2 22 32+ = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1
114. 114. 9(x – 1)2 4(y – 2)2 36 4 36 9 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 (x – 1)2 (y – 2)2 22 32+ = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1 Hence, Center: (1, 2), x-radius is 2, y-radius is 3.
115. 115. 9(x – 1)2 4(y – 2)2 36 4 36 9 Example B. Put 9x2 + 4y2 – 18x – 16y = 11 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 9x2 – 18x + 4y2 – 16y = 11 factor out the square-coefficients 9(x2 – 2x ) + 4(y2 – 4y ) = 11 complete the square 9(x2 – 2x + 1 ) + 4(y2 – 4y + 4 ) = 11 + 9 + 16 +9 +16 + = 1 (x – 1)2 (y – 2)2 22 32+ = 1 Ellipses 9(x – 1)2 + 4(y – 2)2 = 36 divide by 36 to get 1 Hence, Center: (1, 2), x-radius is 2, y-radius is 3. (-1, 2) (3, 2) (1, 5) (1, -1) (1, 2)
116. 116. Hyperbolas
117. 117. Hyperbolas Just as all the other conic sections, hyperbolas are defined by distance relations.
118. 118. Hyperbolas Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. Just as all the other conic sections, hyperbolas are defined by distance relations.
119. 119. A If A, B and C are points on a hyperbola as shown Hyperbolas Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. B C Just as all the other conic sections, hyperbolas are defined by distance relations.
120. 120. A a2 a1 If A, B and C are points on a hyperbola as shown then a1 – a2 Hyperbolas Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. B C Just as all the other conic sections, hyperbolas are defined by distance relations.
121. 121. A a2 a1 b2 b1 If A, B and C are points on a hyperbola as shown then a1 – a2 = b1 – b2 Hyperbolas Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. B C Just as all the other conic sections, hyperbolas are defined by distance relations.
122. 122. A a2 a1 b2 b1 If A, B and C are points on a hyperbola as shown then a1 – a2 = b1 – b2 = c2 – c1 = constant. c1 c2 Hyperbolas Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. B C Just as all the other conic sections, hyperbolas are defined by distance relations.
123. 123. Hyperbolas A hyperbola has a “center”,
124. 124. Hyperbolas A hyperbola has a “center”, and two straight lines that cradle the hyperbolas which are called asymptotes.
125. 125. Hyperbolas A hyperbola has a “center”, and two straight lines that cradle the hyperbolas which are called asymptotes. There are two vertices, one for each branch.
126. 126. Hyperbolas A hyperbola has a “center”, and two straight lines that cradle the hyperbolas which are called asymptotes. There are two vertices, one for each branch. The asymptotes are the diagonals of a box with the vertices of the hyperbola touching the box.
127. 127. Hyperbolas A hyperbola has a “center”, and two straight lines that cradle the hyperbolas which are called asymptotes. There are two vertices, one for each branch. The asymptotes are the diagonals of a box with the vertices of the hyperbola touching the box.
128. 128. Hyperbolas A hyperbola has a “center”, and two straight lines that cradle the hyperbolas which are called asymptotes. There are two vertices, one for each branch. The asymptotes are the diagonals of a box with the vertices of the hyperbola touching the box. The asymptotes are the diagonals of a box with the vertices of the hyperbola touching the box.
129. 129. Hyperbolas The center-box is defined by the x-radius a, and y-radius b as shown. a b
130. 130. Hyperbolas The center-box is defined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. a b
131. 131. Hyperbolas The center-box is defined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are the asymptotes. a b
132. 132. Hyperbolas The center-box is defined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are the asymptotes. Label the vertices and trace the hyperbola along the asympototes. a b
133. 133. Hyperbolas The center-box is defined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are the asymptotes. Label the vertices and trace the hyperbola along the asympototes. a b The location of the center, the x-radius a, and y-radius b may be obtained from the equation.
134. 134. Hyperbolas The equations of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs.
135. 135. Hyperbolas The equations of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs. By completing the square, they may be transformed to the standard forms below.
136. 136. (x – h)2 (y – k)2 a2 b2 Hyperbolas The equations of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs. By completing the square, they may be transformed to the standard forms below. – = 1 (x – h)2(y – k)2 a2b2 – = 1
137. 137. (x – h)2 (y – k)2 a2 b2 (h, k) is the center. Hyperbolas The equations of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs. By completing the square, they may be transformed to the standard forms below. – = 1 (x – h)2(y – k)2 a2b2 – = 1
138. 138. (x – h)2 (y – k)2 a2 b2 x-rad = a, y-rad = b (h, k) is the center. Hyperbolas The equations of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs. By completing the square, they may be transformed to the standard forms below. – = 1 (x – h)2(y – k)2 a2b2 – = 1
139. 139. (x – h)2 (y – k)2 a2 b2 x-rad = a, y-rad = b (h, k) is the center. Hyperbolas The equations of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs. By completing the square, they may be transformed to the standard forms below. – = 1 (x – h)2(y – k)2 a2b2 y-rad = b, x-rad = a – = 1
140. 140. (x – h)2 (y – k)2 a2 b2 x-rad = a, y-rad = b (h, k) is the center. Hyperbolas The equations of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs. By completing the square, they may be transformed to the standard forms below. – = 1 (x – h)2(y – k)2 a2b2 y-rad = b, x-rad = a – = 1 (h, k) Open in the x direction
141. 141. (x – h)2 (y – k)2 a2 b2 x-rad = a, y-rad = b (h, k) is the center. Hyperbolas The equations of hyperbolas have the form Ax2 + By2 + Cx + Dy = E where A and B are opposite signs. By completing the square, they may be transformed to the standard forms below. – = 1 (x – h)2(y – k)2 a2b2 y-rad = b, x-rad = a – = 1 (h, k) Open in the x direction (h, k) Open in the y direction
142. 142. Hyperbolas Following are the steps for graphing a hyperbola.
143. 143. Hyperbolas Following are the steps for graphing a hyperbola. 1. Put the equation into the standard form.
144. 144. Hyperbolas Following are the steps for graphing a hyperbola. 1. Put the equation into the standard form. 2. Read off the center, the x-radius a, the y-radius b, and draw the center-box.
145. 145. Hyperbolas Following are the steps for graphing a hyperbola. 1. Put the equation into the standard form. 2. Read off the center, the x-radius a, the y-radius b, and draw the center-box. 3. Draw the diagonals of the box, which are the asymptotes.
146. 146. Hyperbolas Following are the steps for graphing a hyperbola. 1. Put the equation into the standard form. 2. Read off the center, the x-radius a, the y-radius b, and draw the center-box. 3. Draw the diagonals of the box, which are the asymptotes. 4. Determine the direction of the hyperbolas and label the vertices of the hyperbola.
147. 147. Hyperbolas Following are the steps for graphing a hyperbola. 1. Put the equation into the standard form. 2. Read off the center, the x-radius a, the y-radius b, and draw the center-box. 3. Draw the diagonals of the box, which are the asymptotes. 4. Determine the direction of the hyperbolas and label the vertices of the hyperbola. The vertices are the mid-points of the edges of the center-box.
148. 148. Hyperbolas Following are the steps for graphing a hyperbola. 1. Put the equation into the standard form. 2. Read off the center, the x-radius a, the y-radius b, and draw the center-box. 3. Draw the diagonals of the box, which are the asymptotes. 4. Determine the direction of the hyperbolas and label the vertices of the hyperbola. The vertices are the mid-points of the edges of the center-box. 5. Trace the hyperbola along the asymptotes.
149. 149. Hyperbolas Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1
150. 150. Center: (3, -1) Hyperbolas Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1
151. 151. Center: (3, -1) x-rad = 4 y-rad = 2 Hyperbolas Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1
152. 152. Center: (3, -1) x-rad = 4 y-rad = 2 Hyperbolas (3, -1) 4 2 Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1
153. 153. Center: (3, -1) x-rad = 4 y-rad = 2 Hyperbolas (3, -1) 4 2 Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1
154. 154. Center: (3, -1) x-rad = 4 y-rad = 2 The hyperbola opens left-rt Hyperbolas Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1 (3, -1) 4 2
155. 155. Center: (3, -1) x-rad = 4 y-rad = 2 The hyperbola opens left-rt and the vertices are (7, -1), (-1, -1) . Hyperbolas Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1 (3, -1) 4 2
156. 156. Center: (3, -1) x-rad = 4 y-rad = 2 The hyperbola opens left-rt and the vertices are (7, -1), (-1, -1) . Hyperbolas (3, -1) (7, -1)(-1, -1) 4 2 Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1
157. 157. Center: (3, -1) x-rad = 4 y-rad = 2 The hyperbola opens left-rt and the vertices are (7, -1), (-1, -1) . Hyperbolas (3, -1) (7, -1)(-1, -1) 4 2 Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1
158. 158. Center: (3, -1) x-rad = 4 y-rad = 2 The hyperbola opens left-rt and the vertices are (7, -1), (-1, -1) . Hyperbolas (3, -1) (7, -1)(-1, -1) 4 2 Example A. List the center, the x-radius, the y-radius. Draw the box, the asymptotes, and label the vertices. Trace the hyperbola. (x – 3)2 (y + 1)2 42 22 – = 1 When we use completing the square to get to the standard form of the hyperbolas, because the signs, we add a number and subtract a number from both sides.
159. 159. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Hyperbolas
160. 160. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: Hyperbolas
161. 161. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 Hyperbolas
162. 162. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 Hyperbolas
163. 163. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square Hyperbolas
164. 164. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x ) = 29 Hyperbolas
165. 165. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 Hyperbolas
166. 166. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 16 Hyperbolas
167. 167. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 16 –9 Hyperbolas
168. 168. Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas
169. 169. 4(y – 2)2 – 9(x + 1)2 = 36 Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas
170. 170. 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas
171. 171. 9(x + 1)24(y – 2)2 36 36 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 – = 1 Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas
172. 172. 9(x + 1)24(y – 2)2 36 36 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 – = 1 Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas 9 4
173. 173. 9(x + 1)24(y – 2)2 36 36 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 – = 1 Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas (y – 2)2 (x + 1)2 32 22 – = 1 9 4
174. 174. 9(x + 1)24(y – 2)2 36 36 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 – = 1 Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas (y – 2)2 (x + 1)2 32 22 – = 1 Center: (-1, 2), 9 4
175. 175. 9(x + 1)24(y – 2)2 36 36 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 – = 1 Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas (y – 2)2 (x + 1)2 32 22 – = 1 Center: (-1, 2), x-rad = 2, y-rad = 3 9 4
176. 176. 9(x + 1)24(y – 2)2 36 36 4(y – 2)2 – 9(x + 1)2 = 36 divide by 36 to get 1 – = 1 Example B. Put 4y2 – 9x2 – 18x – 16y = 29 into the standard form. Find the center, major and minor axis. Draw and label the top, bottom, right, left most points. Group the x’s and the y’s: 4y2 – 16y – 9x2 – 18x = 29 factor out the square-coefficients 4(y2 – 4y ) – 9(x2 + 2x ) = 29 complete the square 4(y2 – 4y + 4 ) – 9(x2 + 2x + 1 ) = 29 + 16 – 9 16 –9 Hyperbolas (y – 2)2 (x + 1)2 32 22 – = 1 Center: (-1, 2), x-rad = 2, y-rad = 3 The hyperbola opens up and down. 9 4
177. 177. (-1, 2) Hyperbolas Center: (-1, 2), x-rad = 2, y-rad = 3
178. 178. (-1, 2) (-1, 5) (-1, -1) Hyperbolas Center: (-1, 2), x-rad = 2, y-rad = 3 The hyperbola opens up and down. The vertices are (-1, -1) and (-1, 5).
179. 179. (-1, 2) (-1, 5) (-1, -1) Hyperbolas Center: (-1, 2), x-rad = 2, y-rad = 3 The hyperbola opens up and down. The vertices are (-1, -1) and (-1, 5).
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