32 conic sections, circles and completing the square
- 2. Conic Sections One way to study a solid is to slice it open.
- 3. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area.
- 4. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A right circular cone
- 5. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A right circular cone and conic sections (wikipedia “Conic Sections”)
- 6. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A Horizontal Section A right circular cone and conic sections (wikipedia “Conic Sections”)
- 7. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A Horizontal Section A right circular cone and conic sections (wikipedia “Conic Sections”)
- 8. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A Moderately Tilted Section A right circular cone and conic sections (wikipedia “Conic Sections”)
- 9. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A Moderately Tilted Section A right circular cone and conic sections (wikipedia “Conic Sections”)
- 10. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A Horizontal Section A Moderately Tilted Section Circles and ellipsis are enclosed. A right circular cone and conic sections (wikipedia “Conic Sections”)
- 11. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A Parallel–Section A right circular cone and conic sections (wikipedia “Conic Sections”)
- 12. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A Parallel–Section A right circular cone and conic sections (wikipedia “Conic Sections”)
- 13. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. An Cut-away Section A right circular cone and conic sections (wikipedia “Conic Sections”)
- 14. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. An Cut-away Section A right circular cone and conic sections (wikipedia “Conic Sections”)
- 15. Conic Sections One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown. A Horizontal Section A Moderately A Parallel–Section Tilted Section An Cut-away Circles and Section ellipsis are enclosed. Parabolas and hyperbolas are open. A right circular cone and conic sections (wikipedia “Conic Sections”)
- 16. Conic Sections We summarize the four types of conics sections here. Circles Ellipses Parabolas Hyperbolas
- 17. Conic Sections We summarize the four types of conics sections here. Circles Ellipses Parabolas Hyperbolas Besides their differences in visual appearance and the manners they reside inside the cone, there are many reasons, that have nothing to do with cones, that the conic sections are grouped into four groups.
- 18. Conic Sections We summarize the four types of conics sections here. Circles Ellipses Parabolas Hyperbolas Besides their differences in visual appearance and the manners they reside inside the cone, there are many reasons, that have nothing to do with cones, that the conic sections are grouped into four groups. One way is to use distance relations to classify them.
- 19. Conic Sections We summarize the four types of conics sections here. Circles Ellipses Parabolas Hyperbolas Besides their differences in visual appearance and the manners they reside inside the cone, there are many reasons, that have nothing to do with cones, that the conic sections are grouped into four groups. One way is to use distance relations to classify them. We use the circles and the ellipsis as examples.
- 20. Circles Given a fixed point C, C
- 21. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. C
- 22. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. C
- 23. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. C
- 24. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. C Hence a dog tied to a post would mark off a circular track.
- 25. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle.
- 26. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle. Given two fixed points (called foci), F1 F2
- 27. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle. Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. F1 F2
- 28. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle. Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. Q For example, if P, Q, and R P are points on a ellipse, F1 F2 R
- 29. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle. Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. Q For example, if P, Q, and R P are points on a ellipse, then p2 p1 p1 + p2 F1 F2 R
- 30. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle. Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. Q For example, if P, Q, and R P p2 q1 are points on a ellipse, then q2 p1 p1 + p2 F1 F2 = q1 + q2 R
- 31. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle. Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. Q For example, if P, Q, and R P p2 q1 are points on a ellipse, then q2 p1 p1 + p2 F1 F2 = q1 + q2 = r1 + r 2 r1 r2 = a constant R
- 32. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle. Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. Q For example, if P, Q, and R P p2 q1 are points on a ellipse, then q2 p1 p1 + p2 F1 F2 = q1 + q2 = r1 + r 2 r1 r2 = a constant Hence a dog leashed by a ring R to two posts would mark off an elliptical track.
- 33. Circles Given a fixed point C, a circle is the set of points whose distances to C is r r a fixed constant r. The equal-distance r is called the C radius and the point C is called the center of the circle. Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant. Q For example, if P, Q, and R P p2 q1 are points on a ellipse, then q2 p1 p1 + p2 F1 F2 = q1 + q2 = r1 + r 2 r1 r2 = a constant Likewise parabolas and hyperbolas R may be defined using relations of distance measurements.
- 34. Conic Sections The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types
- 35. Conic Sections The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types Recall that straight lines are the graphs of 1st degree equations Ax + By = C where A, B, C, are numbers.
- 36. Conic Sections The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types Recall that straight lines are the graphs of 1st degree equations Ax + By = C y = –1 x=1 y+x=1 Linear graphs where A, B, C, are numbers.
- 37. Conic Sections The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types Recall that straight lines are the graphs of 1st degree equations Ax + By = C y = –1 x=1 y+x=1 Linear graphs where A, B, C, are numbers. Conic sections are the graphs of 2nd degree equations in x and y.
- 38. Conic Sections The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types Recall that straight lines are the graphs of 1st degree equations Ax + By = C y = –1 x=1 y+x=1 Linear graphs where A, B, C, are numbers. 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.
- 39. Conic Sections The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types Recall that straight lines are the graphs of 1st degree equations Ax + By = C y = –1 x=1 y+x=1 Linear graphs where A, B, C, are numbers. 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. The algebraic technique that enables us to sort these 2nd degree equations into four groups of conic sections is called "completing the square".
- 40. Conic Sections The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types Recall that straight lines are the graphs of 1st degree equations Ax + By = C y = –1 x=1 y+x=1 Linear graphs where A, B, C, are numbers. 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. The algebraic technique that enables us to sort these 2nd degree equations into four groups of conic sections is called "completing the square". We will apply this method to the circles but only summarize the results about the other ones.
- 41. 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. r r center
- 42. 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. The radius and the center completely determine the circle. r r center
- 43. 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. The radius and the center completely determine the circle. Let (h, k) be the center of a circle and r be the radius. (h, k) r
- 44. 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. The radius and the center completely determine the circle. Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on (x, y) the circle, then the distance between (x, y) and the center (h, k) r is r.
- 45. 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. The radius and the center completely determine the circle. Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on (x, y) the circle, then the distance between (x, y) and the center (h, k) r is r. Hence, r = √ (x – h)2 + (y – k)2
- 46. 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. The radius and the center completely determine the circle. Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on (x, y) the circle, then the distance between (x, y) and the center (h, k) r is r. Hence, r = √ (x – h)2 + (y – k)2 or r2 = (x – h)2 + (y – k)2
- 47. 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. The radius and the center completely determine the circle. Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on (x, y) the circle, then the distance between (x, y) and the center (h, k) r is r. Hence, r = √ (x – h)2 + (y – k)2 or r2 = (x – h)2 + (y – k)2 This is called the standard form of circles.
- 48. 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. The radius and the center completely determine the circle. Let (h, k) be the center of a circle and r be the radius. Suppose (x, y) is a point on (x, y) the circle, then the distance between (x, y) and the center (h, k) r is r. Hence, 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.
- 49. Circles r2 = (x – h)2 + (y – k)2
- 50. Circles must be “ – ” r2 = (x – h)2 + (y – k)2
- 51. Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2
- 52. Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the center
- 53. Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the center Example A. Write the equation (–1, 8) of the circle as shown. (–1, 3)
- 54. Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the center Example A. Write the equation (–1, 8) of the circle as shown. The center is (–1, 3) and the radius is 5. (–1, 3)
- 55. Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the center Example A. Write the equation (–1, 8) of the circle as shown. The center is (–1, 3) and the radius is 5. (–1, 3) Hence the equation is: 52 = (x – (–1))2 + (y – 3)2
- 56. Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the center Example A. Write the equation (–1, 8) of the circle as shown. The center is (–1, 3) and the radius is 5. (–1, 3) Hence the equation is: 52 = (x – (–1))2 + (y – 3)2 or 25 = (x + 1)2 + (y – 3 )2
- 57. Circles r is the radius must be “ – ” r2 = (x – h)2 + (y – k)2 (h, k) is the center Example A. Write the equation (–1, 8) of the circle as shown. The center is (–1, 3) and the radius is 5. (–1, 3) Hence the equation is: 52 = (x – (–1))2 + (y – 3)2 or 25 = (x + 1)2 + (y – 3 )2 In particular a circle centered at the origin has an equation of the form x2 + y2 = r2
- 58. Circles Example B. 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.
- 59. Circles Example B. 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
- 60. Circles Example B. 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)
- 61. Circles Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. (3, 2) Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2) standard form: (3, --2) 42 = (x – 3)2 + (y – (–2))2 (3, --6) Hence r = 4, center = (3, –2)
- 62. Circles Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. (3, 2) Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2) standard form: (3, --2) 42 = (x – 3)2 + (y – (–2))2 (3, --6) Hence r = 4, center = (3, –2) When equations are not in the standard form, we have to rearrange them into the standard form. We do this by "completing the square".
- 63. Circles Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. (3, 2) Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2) standard form: (3, --2) 42 = (x – 3)2 + (y – (–2))2 (3, --6) Hence r = 4, center = (3, –2) 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.
- 64. Circles Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. (3, 2) Label the top, bottom, left and right most points. Graph it. Put 16 = (x – 3)2 + (y + 2)2 into the (--1, --2) (7, --2) standard form: (3, --2) 42 = (x – 3)2 + (y – (–2))2 (3, --6) Hence r = 4, center = (3, –2) 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.
- 65. Circles The Completing the Square Method
- 66. Circles The Completing the Square Method If we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square,
- 67. Circles The Completing the Square Method 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.
- 68. Circles The Completing the Square Method 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. Example C. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2
- 69. Circles The Completing the Square Method 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. Example C. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2
- 70. Circles The Completing the Square Method 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. Example C. Fill in the blank to make a perfect square. a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
- 71. Circles The Completing the Square Method 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. Example C. 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
- 72. Circles The Completing the Square Method 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. Example C. 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
- 73. Circles The Completing the Square Method 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. Example C. 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
- 74. Circles The Completing the Square Method 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. Example C. 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.
- 75. Circles The Completing the Square Method 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. Example C. 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 to the other side of the equation.
- 76. Circles The Completing the Square Method 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. Example C. 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 to the other side of the equation. 2. Complete the square for the x-terms and for the y-terms. Make sure to add the necessary numbers to both sides.
- 77. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
- 78. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form:
- 79. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36
- 80. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
- 81. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
- 82. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9
- 83. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32
- 84. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the 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.
- 85. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the 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. (3, –6),
- 86. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the 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), (3, –3), and radius is 3. (0, –6), (6, –6), (3, –6), (–9, –6)
- 87. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the 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), (3, –3), and radius is 3. The Completing-the-Square method is the basic method for (0, –6), (6, –6), (3, –6), handling 2nd degree problems. (–9, –6)
- 88. Circles Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it. We use completing the square to put the equation into the standard form: x2 – 6x + + y2 + 12y + = –36 complete the 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), (3, –3), and radius is 3. The Completing-the-Square method is the basic method for (0, –6), (6, –6), (3, –6), handling 2nd degree problems. We summarize the hyperbola and parabola below. (–9, –6)
- 89. Hyperbolas
- 90. Hyperbolas Just as all the other conic sections, hyperbolas are defined by distance relations.
- 91. Hyperbolas Just as all the other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant.
- 92. Hyperbolas Just as all the other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. If A, B and C are points on a hyperbola as shown C A B
- 93. Hyperbolas Just as all the other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. If A, B and C are points on a hyperbola as shown then a 1 – a2 C A a1 a2 B
- 94. Hyperbolas Just as all the other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. If A, B and C are points on a hyperbola as shown then a1 – a2 = b1 – b2 C A a1 a2 b2 B b1
- 95. Hyperbolas Just as all the other conic sections, hyperbolas are defined by distance relations. Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant. If A, B and C are points on a hyperbola as shown then a1 – a2 = b1 – b2 = c2 – c1 = constant. C c2 A a1 c1 a2 b2 B b1
- 96. Parabolas Finally, we illustrate the definition that’s based on distance measurements of the parabolas. Given a fixed point F, and a line L, the points that are of equal distance from F the line L is a parabola. Hence a = A, b = B, c = C as shown below. For more information, see: http://en.wikipedia.org/wiki/Parabola F L
- 97. Parabolas Finally, we illustrate the definition that’s based on distance measurements of the parabolas. Given a fixed point F, and a line L, the points that are of equal distance from F the line L is a parabola. Hence a = A, b = B, c = C as shown below. For more information, see: http://en.wikipedia.org/wiki/Parabola P1 F a A L
- 98. Parabolas Finally, we illustrate the definition that’s based on distance measurements of the parabolas. Given a fixed point F, and a line L, the points that are of equal distance from F the line L is a parabola. Hence a = A, b = B, c = C as shown below. For more information, see: http://en.wikipedia.org/wiki/Parabola P1 F a b P2 A B L
- 99. Parabolas Finally, we illustrate the definition that’s based on distance measurements of the parabolas. Given a fixed point F, and a line L, the points that are of equal distance from F the line L is a parabola. Hence a = A, b = B, c = C as shown below. For more information, see: http://en.wikipedia.org/wiki/Parabola a b c A B C