©Mathworld

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

Conic Sections - Four Types

© Art Mayoff
© Long Island Fountain Company

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 © 1997-2004, Math Academy Online™ / Platonic Realms™.

© Jill Britton, September 25, 2003

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.

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.

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.

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.

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.

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).

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.

Parabola
• The Standard Form of a Parabola that opens to the right and has a vertex at (0,0) is…… y
2 
4 px
©1999 Addison Wesley Longman, Inc.

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)

Parabola
• The Standard Form of a Parabola that opens to the left and has a vertex at (0,0) is…… y
2  
4 px
© Shelly Walsh

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)

Parabola
• The Standard Form of a Parabola that opens up and has a vertex at (0,0) is…… x
2 
4 py
©1999-2003 SparkNotes LLC, All Rights Reserved

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)

Parabola
• The Standard Form of a Parabola that opens down and has a vertex at (0,0) is…… x
2  
4 py
©1999 Addison Wesley Longman, Inc.

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)

Parabola
• The Standard Form of a Parabola that opens to the right and has a vertex at (h,k) is……
( y
 k )
2 
4 p ( x
 h )
© Shelly Walsh

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
 b y

2 a

Parabola
• The Standard Form of a Parabola that opens to the left and has a vertex at (h,k) is……
( y
 k )
2  
4 p ( x
 h )
©June Jones, University of Georgia

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  b y

2 a

Parabola
• The Standard Form of a Parabola that opens up and has a vertex at (h,k) is……
( x
 h )
2 
4 p ( y
 k )
Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

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 x

 b
2 a

Parabola
• The Standard Form of a Parabola that opens down and has a vertex at (h,k) is……
( x
 h )
2  
4 p ( y
 k )
Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

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 x

 b
2 a

© 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.

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 © 1997-2004, Math Academy Online™ / Platonic Realms™.

• 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.

• 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.
© 1994-2004 Kevin Matthews and
Artifice, Inc.
All Rights Reserved.

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.

The Ellipse
• Any cylinder sliced on an angle will reveal an ellipse in crosssection (as seen in the Tycho Brahe
Planetarium in
Copenhagen).

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.

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.

The Ellipse

The Ellipse
• 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 1759. 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

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

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.

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.

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 eavesdropping on the private conversations of other
House members located near the other focal point.

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.

Ellipse
•
General Rules
– x and y are both squared
– Equation always equals(=) 1
– Equation is always plus(+)
– a 2 is always the biggest denominator
– c 2 = a 2 – b 2
– 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.

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)

Ellipse
• The standard form of the ellipse with a center at (0,0) and a horizontal axis is…… x
2 a
2
 y
2 b
2

1

Ellipse
• The ellipse with a center at (0,0) and a horizontal axis has the following characteristics…… x
2
16
 y
2
9

1
© Cabalbag, Porter, Chadwick, and Liefting

Ellipse
• The standard form of the ellipse with a center at (0,0) and a vertical axis is…… x
2 b
2
 y
2 a
2

1

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) x
2
9 y
2

81

1
© Cabalbag, Porter, Chadwick, and Liefting

Ellipse
• The standard form of the ellipse with a center at (h,k) and a horizontal axis is……
( x
 h )
a

( y
 k )
b

1

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

Ellipse
• The standard form of the ellipse with a center at
(h,k) and a vertical axis is……
( x
 h )
2 b
2

( y
 k )
2 a
2

1

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 1998 -2000

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.

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 © 1997-2004, Math Academy Online™ / Platonic Realms™.

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

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.

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.

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.

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 ( L ong
R ange N avigation).

Hyperbola
• General Rules
– x and y are both squared
– Equation always equals(=) 1
– Equation is always minus(-)
– a 2 is always the first denominator
– c 2 = a 2 + b 2
– 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

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 x 2 is first then the hyperbola is horizontal
– If y 2 is first then the hyperbola is vertical.

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

Hyperbola
• The standard form of the Hyperbola with a center at
(0,0) and a horizontal axis is…… x
2 a
2
 y
2 b
2

1

Hyperbola
• The Hyperbola with a center at (0,0) and a horizontal axis has the following characteristics……
• Vertices ( a,0)
• Foci ( c,0)
• Asymptotes: y
  b a x

Hyperbola
• The standard form of the Hyperbola with a center at
(0,0) and a vertical axis is…… y
2 a
2
 x
2 b
2


Hyperbola
• The Hyperbola with a center at (0,0) and a vertical axis has the following characteristics……
• Vertices (0, a)
•
Foci ( 0, c)
•
Asymptotes: y
  a b x

Hyperbola
• The standard form of the Hyperbola with a center at
(h,k) and a horizontal axis is……
( x
 h )
2 a
2

( y
 k )
2 b
2

1

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: y
 k
 b a
( x
 h )

Hyperbola
• The standard form of the Hyperbola with a center at
(h,k) and a vertical axis is……
( y
 k )
2 a
2

( x
 h )
2 b
2

1

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: y
 k
 a b
( x
 h )
©Sellers, James

Rotating the Coordinate Axis
Ax
2 
Bxy

Cy
2 
Dx

Ey

F

0
© James Wilson

x
 x ' cos
  y ' sin
 y
 x ' sin
  y ' cos
 cot 2
 
A

C
B or tan 2
 
B
A

C

Resources
Bookbinder, John. Unit 8: Conic Sections (College Algebra
Online). 2000. June 3, 2004
<http://www.distancemath.com/unit8/ch8p1.htm>.
Britton, Jill. Occurrence of the Conics. September 25, 2003. June 3,
2004 <http://ccins.camosun.bc.ca/~jbritton/jbconics.htm>.
Cabalbag, Christain, and Porter, Amanda and Chadwick, Justin and
Liefting. Nick. Graphing Conic Sections (Microsoft Power Point
Presentation 1997). 2001. June3, 2004
<http://www.granite.k12.ut.us/Hunter_High/StaffPages/Olsen_P/Cla ssWebSite/2003%20student%20projects/27circlesandelipse.ppt

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, 2004 http://jwilson.coe.uga.edu/emt669/Student.Folders/Jones.June/conic s/conics.html
Mathews, Kevin. Great Buildings Online. Great Buildings. une 3,
2004 <http://www.GreatBuildings.com/buildings/Saint_Pauls_Cathe dral.html

Resources
Mayoff, Art. San Francisco and the Golden Gate Bridge.
June 3, 2004 http://mathworld.wolfram.com/ConicSection.html>.
Mueller, William. Modeling Periodicity .
June 3, 2004
<http://www.wmueller.com/precalculus/funcdata/1_10.html>.
PRIME Articles. Platomic Realms.
June 3, 2004
<http://www.mathacademy.com/pr/prime/index.asp>.

Resources
Quadratics. Spark Notes from Barnes and Noble.
June 3, 2004
<http://www.sparknotes.com/math/algebra1/quadratics/section1.html
Roberts, Donna. Mathematics A . Oswego City School District
Regents Exam Prep.
June, 3, 2004 <http://regentsprep.org/Regents/math/mathtopic.cfm?TopicCode=conics>.
Seek One Web Services, Long Island Fountain
Company. <http://www.lifountain.com/fountainideas.html>.
Sellers, James, Introduction to Conics, June 8, 2004. http://www.krellinst.org/UCES/archive/resources/conics/newconics.ht
ml

Resources
Walsh, Shelly. Chapter 9 (Precalculus).
June 3, 2004 http://faculty.ed.umuc.edu/~swalsh/UM/M108Ch9.html
Weissteing, Eric W. "Conic Section." From MathWorld --A Wolfram Web
Resource. http://mathworld.wolfram.com/ConicSection.html
Wilson, James W. CURVE BUILDING. An Exploration with
Algebraic Relations University of Georgia.
June 3, 2004 http://jwilson.coe.uga.edu/Texts.Folder/cb/curve.building.html

Other Works Cited
• Jill Britton
– http://britton.disted.camosun.bc.ca/jbconics.ht
m
• Online Encyclopedia
– Wikipedia
• The Comet’s Tale: Orbits
– http://cse.ssl.berkeley.edu/SegwayEd/lessons
/cometstale/frame_orbits.html

©National Science Foundation

Circle
• The Standard Form of a circle with a center at (0,0) and a radius, r, is……..
2 x
 y
2  r
2 center (0,0) radius = 2
Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

(
Circles
• The Standard Form of a circle with a center at (h,k) and a radius, r, is……..
x
 h )
2 
( y
 k )
2  r
2
Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center center (3,3) radius = 2