1. Conic Sections
©Mathworld

2. Conic Sections - Definition
A conic section is a curve formed by intersecting cone with a plane
There are four types of Conic sections

3. Conic Sections - Four Types

4. Parabolas
© Art Mayoff
© Long Island Fountain Company

5. What’s in a Parabola
A parabola is the set of all points in a plane such that each point in the set is equidistant from a line called the directrix and a fixed point called the focus.
(x, y)
Copyright © , Math Academy Online™ / Platonic Realms™.

6. Why is the focus so important?
© Jill Britton, September 25, 2003

7. The Parabola
One of nature's best known approximations to parabolas is the path taken by a body projected upward and obliquely to the pull of gravity, as in the parabolic trajectory of a golf ball. The friction of air and the pull of gravity will change slightly the projectile's path from that of a true parabola, but in many cases the error is insignificant.

8. The Parabola
This discovery by Galileo in the 17th century made it possible for cannoneers to work out the kind of path a cannonball would travel if it were hurtled through the air at a specific angle.

9. The Parabola
When a baseball is hit into the air, it follows a parabolic path; the center of gravity of a leaping porpoise describes a parabola.

10. The Parabola
The easiest way to visualize the path of a projectile is to observe a waterspout. Each molecule of water follows the same path and, therefore, reveals a picture of the curve. The fountains of the Bellagio Hotel in Las Vegas comprise a parabolic chorus line.

11. The Parabola
Parabolas exhibit unusual and useful reflective properties. If a light is placed at the focus of a parabolic mirror (a curved surface formed by rotating a parabola about its axis), the light will be reflected in rays parallel to said axis. In this way a straight beam of light is formed. It is for this reason that parabolic surfaces are used for headlamp reflectors. The bulb is placed at the focus for the high beam and a little above the focus for the low beam.

12. The Parabola
The opposite principle is used in the giant mirrors in reflecting telescopes and in antennas used to collect light and radio waves from outer space: the beam comes toward the parabolic surface and is brought into focus at the focal point. The instrument with the largest single-piece parabolic mirror is the Subaru telescope at the summit of Mauna Kea in Hawaii (effective diameter: 8.2 m).

13. The Parabola
Heat waves, as well as light and sound waves, are reflected to the focal point of a parabolic surface. If a parabolic reflector is turned toward the sun, flammable material placed at the focus may ignite. A solar furnace produces heat by focusing sunlight by means of a parabolic mirror arrangement. Light is sent to it by set of moveable mirrors computerized to follow the sun during the day. Solar cooking involves a similar use of a parabolic mirror.

14. Circle
©National Science Foundation

15. Circle
The Standard Form of a circle with a center at (0,0) and a radius, r, is……..
                                                                     center (0,0) radius = 2
Copyright © Oswego City School District Regents Exam Prep Center

16. Circles
The Standard Form of a circle with a center at (h,k) and a radius, r, is……..
                                                                                                                                                                          center (3,3) radius = 2
Copyright © Oswego City School District Regents Exam Prep Center

17. Parabola
The Standard Form of a Parabola that opens to the right and has a vertex at (0,0) is……
©1999 Addison Wesley Longman, Inc.

18. Parabola
The Parabola that opens to the right and has a vertex at (0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (p,0)
This makes the equation of the directrix x = -p
The makes the axis of symmetry the x-axis (y = 0)

19. Parabola
The Standard Form of a Parabola that opens to the left and has a vertex at (0,0) is……
© Shelly Walsh

20. Parabola
The Parabola that opens to the left and has a vertex at (0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus(-p,0)
This makes the equation of the directrix x = p
The makes the axis of symmetry the x-axis (y = 0)

21. Parabola
The Standard Form of a Parabola that opens up and has a vertex at (0,0) is……
© SparkNotes LLC, All Rights Reserved

22. Parabola
The Parabola that opens up and has a vertex at (0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (0,p)
This makes the equation of the directrix y = -p
This makes the axis of symmetry the y-axis (x = 0)

23. Parabola
The Standard Form of a Parabola that opens down and has a vertex at (0,0) is……
©1999 Addison Wesley Longman, Inc.

24. Parabola
The Parabola that opens down and has a vertex at (0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (0,-p)
This makes the equation of the directrix y = p
This makes the axis of symmetry the y-axis (x = 0)

25. Parabola
The Standard Form of a Parabola that opens to the right and has a vertex at (h,k) is……
© Shelly Walsh

26. Parabola
The Parabola that opens to the right and has a vertex at (h,k) has the following characteristics……..
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (h+p, k)
This makes the equation of the directrix x = h – p
This makes the axis of symmetry

27. Parabola
The Standard Form of a Parabola that opens to the left and has a vertex at (h,k) is……
©June Jones, University of Georgia

28. Parabola
The Parabola that opens to the left and has a vertex at (h,k) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (h – p, k)
This makes the equation of the directrix x = h + p
The makes the axis of symmetry

29. Parabola
The Standard Form of a Parabola that opens up and has a vertex at (h,k) is……
Copyright © Oswego City School District Regents Exam Prep Center

30. Parabola
The Parabola that opens up and has a vertex at (h,k) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (h , k + p)
This makes the equation of the directrix y = k – p
The makes the axis of symmetry

31. Parabola
The Standard Form of a Parabola that opens down and has a vertex at (h,k) is……
Copyright © Oswego City School District Regents Exam Prep Center

32. Parabola
The Parabola that opens down and has a vertex at (h,k) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (h , k - p)
This makes the equation of the directrix y = k + p
This makes the axis of symmetry

33. Ellipse © Jill Britton, September 25, 2003
Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point.

34. What is in an Ellipse?
The set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant. (“Foci” is the plural of “focus”, and is pronounced FOH-sigh.)
Copyright © , Math Academy Online™ / Platonic Realms™.

35. Why are the foci of the ellipse important?
The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

36. Why are the foci of the ellipse important?
St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.
© Kevin Matthews and Artifice, Inc. All Rights Reserved.

37. The Ellipse
Though not so simple as the circle, the ellipse is nevertheless the curve most often "seen" in everyday life. The reason is that every circle, viewed obliquely, appears elliptical.

38. The Ellipse
Any cylinder sliced on an angle will reveal an ellipse in cross-section (as seen in the Tycho Brahe Planetarium in Copenhagen).

39. The Ellipse
Tilt a glass of water and the surface of the liquid acquires an elliptical outline. Salami is often cut obliquely to obtain elliptical slices which are larger.

40. The Ellipse
The early Greek astronomers thought that the planets moved in circular orbits about an unmoving earth, since the circle is the simplest mathematical curve. In the 17th century, Johannes Kepler eventually discovered that each planet travels around the sun in an elliptical orbit with the sun at one of its foci.

41. The Ellipse

42. The Ellipse Click here to see the orbit
The orbits of the moon and of artificial satellites of the earth are also elliptical as are the paths of comets in permanent orbit around the sun.
Halley's Comet takes about 76 years to travel abound our sun. Edmund Halley saw the comet in 1682 and correctly predicted its return in Although he did not live long enough to see his prediction come true, the comet is named in his honor.
Click here to see the orbit

43. The Ellipse
On a far smaller scale, the electrons of an atom move in an approximately elliptical orbit with the nucleus at one focus.

44. The Ellipse
The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

45. The Ellipse
The principle is also used in the construction of "whispering galleries" such as in St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.

46. The Ellipse
Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eaves-dropping on the private conversations of other House members located near the other focal point.

47. The Ellipse
The ability of the ellipse to rebound an object starting from one focus to the other focus can be demonstrated with an elliptical billiard table. When a ball is placed at one focus and is thrust with a cue stick, it will rebound to the other focus. If the billiard table is live enough, the ball will continue passing through each focus and rebound to the other.

48. Ellipse General Rules x and y are both squared
Equation always equals(=) 1
Equation is always plus(+) in middle
a2 is always the biggest denominator
c2 = a2 – b2
c is the distance from the center to each foci on the major axis
The center is in the middle of the 2 vertices, the 2 covertices, and the 2 foci.

49. Ellipse
General Rules
a is the distance from the center to each vertex on the major axis
b is the distance from the center to each vertex on the minor axis (co-vertices)
Major axis has a length of 2a
Minor axis has a length of 2b
Eccentricity(e): e = c/a (The closer e gets to 0, the closer it is to being circular)

50. A note about Eccentricity
A number that indicates how drawn out or attenuated a conic section is. Eccentricity is represented by the letter e (no relation to e = )
eccentricity
Conic Section
e = 0
Circle
0 < e < 1
Ellipse
e = 1
Parabola
e > 1
Hyperbola
10.2

51. Ellipse
The standard form of the ellipse with a center at (0,0) and a horizontal axis is……

52. Ellipse
The ellipse with a center at (0,0) and a horizontal axis has the following characteristics……
Vertices ( a,0)
Co-Vertices (0, b)
Foci ( c,0)
© Cabalbag, Porter, Chadwick, and Liefting

53. Ellipse
The standard form of the ellipse with a center at (0,0) and a vertical axis is……

54. Ellipse
The ellipse with a center at (0,0) and a vertical axis has the following characteristics……
Vertices (0, a)
Co-Vertices ( b,0)
Foci (0, c)
© Cabalbag, Porter, Chadwick, and Liefting

55. Ellipse
The standard form of the ellipse with a center at (h,k) and a horizontal axis is……

56. Ellipse
The ellipse with a center at (h,k) and a horizontal axis has the following characteristics……
Vertices (h a , k)
Co-Vertices (h, k b)
Foci (h c , k)
©Sellers, James

57. Ellipse
The standard form of the ellipse with a center at (h,k) and a vertical axis is……

58. Ellipse
The ellipse with a center at (h,k) and a vertical axis has the following characteristics……
Vertices (h, k a)
Co-Vertices (h b , k)
Foci (h, k c)
© Joan Bookbinder

59. Hyperbola © Jill Britton, September 25, 2003
The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does
the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

60. What is a Hyperbola?
The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
Copyright © , Math Academy Online™ / Platonic Realms™.

61. Where are the Hyperbolas?
A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.
© Jill Britton, September 25, 2003

62. The Hyperbola
If a right circular cone is intersected by a plane parallel to its axis, part of a hyperbola is formed. Such an intersection can occur in physical situations as simple as sharpening a pencil that has a polygonal cross section or in the patterns formed on a wall by a lamp shade.

63. The Hyperbola
A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.

64. The Hyperbola
A hyperbola revolving around its axis forms a surface called a hyperboloid. The cooling tower of a steam power plant has the shape of a hyperboloid, as does the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

65. The Hyperbola
All three conic sections can be characterized by moiré patterns. If the center of each of two sets of concentric circles is the source of a radio signal, the synchronized signals would intersect one another in associated hyperbolas. This principle forms the basis of a hyperbolic radio navigation system known as Loran (Long Range Navigation).

66. Hyperbola General Rules x and y are both squared
Equation always equals(=) 1
Equation is always minus(-)
a2 is always the first denominator
c2 = a2 + b2
c is the distance from the center to each foci on the major axis
a is the distance from the center to each vertex on the major axis

67. Hyperbola General Rules
b is the distance from the center to each midpoint of the rectangle used to draw the asymptotes. This distance runs perpendicular to the distance (a).
Major axis has a length of 2a
Eccentricity(e): e = c/a (The closer e gets to 1, the closer it is to being circular
If x2 is first then the hyperbola is horizontal
If y2 is first then the hyperbola is vertical.

68. Hyperbola General Rules
The center is in the middle of the 2 vertices and the 2 foci.
The vertices and the covertices are used to draw the rectangles that form the asymptotes.
The vertices and the covertices are the midpoints of the rectangle
The covertices are not labeled on the hyperbola because they are not actually part of the graph

69. Hyperbola
The standard form of the Hyperbola with a center at (0,0) and a horizontal axis is……

70. Hyperbola
The Hyperbola with a center at (0,0) and a horizontal axis has the following characteristics……
Vertices ( a,0)
Foci ( c,0)
Asymptotes:

71. Hyperbola
The standard form of the Hyperbola with a center at (0,0) and a vertical axis is……

72. Hyperbola
The Hyperbola with a center at (0,0) and a vertical axis has the following characteristics……
Vertices (0, a)
Foci ( 0, c)
Asymptotes:

73. Hyperbola
The standard form of the Hyperbola with a center at (h,k) and a horizontal axis is……

74. Hyperbola
The Hyperbola with a center at (h,k) and a horizontal axis has the following characteristics……
Vertices (h a, k)
Foci (h c, k )
Asymptotes:

75. Hyperbola
The standard form of the Hyperbola with a center at (h,k) and a vertical axis is……

76. Hyperbola
The Hyperbola with a center at (h,k) and a vertical axis has the following characteristics……
Vertices (h, k a)
Foci (h, k c)
Asymptotes:
©Sellers, James

77. Rotating the Coordinate Axis
© James Wilson

78. Equations for Rotating the Coordinate Axes

79. Resources
Bookbinder, John. Unit 8: Conic Sections (College Algebra Online). 2000. June 3, 2004 <
Britton, Jill. Occurrence of the Conics. September 25, 2003. June 3, 2004 <
Cabalbag, Christain, and Porter, Amanda and Chadwick, Justin and Liefting. Nick. Graphing Conic Sections (Microsoft Power Point Presentation 1997).  June3, 2004 <

80. Resources
Finney, Ross, et. al. Calculus: Graphical, Numerical, Algebraic. Scott Foresman-Addison Wesley, 1999.
Jones, June. Instructional Unit on Conic Sections. University of Georgia. 
June 3,
Mathews, Kevin. Great Buildings Online. Great Buildings. une 3, 2004 <

81. Resources Mueller, William. Modeling Periodicity .
Mayoff, Art. San Francisco and the Golden Gate Bridge. 
June 3, 2004 
Mueller, William. Modeling Periodicity . 
<
PRIME Articles. Platomic Realms. 
<

82. Resources
Quadratics. Spark Notes from Barnes and Noble. 
June 3, 2004 
<
Roberts, Donna. Mathematics A . Oswego City School District Regents Exam Prep. 
June, 3, 2004 <
Seek One Web Services, Long Island Fountain Company. <
Sellers, James, Introduction to Conics, June 8, 2004.

83. Resources Walsh, Shelly. Chapter 9 (Precalculus). June 3, 2004
   
Walsh, Shelly. Chapter 9 (Precalculus). 
June 3, 2004 
Weissteing, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource.
Wilson, James W.  CURVE BUILDING. An Exploration with Algebraic Relations University of Georgia. 
June 3,

84. Other Works Cited Jill Britton Online Encyclopedia
Online Encyclopedia
Wikipedia
The Comet’s Tale: Orbits