Conic Sections ©Mathworld Circle ©National Science Foundation.

Conic Sections

©Mathworld

Circle

©National Science Foundation

Circle

• The Standard Form of a circle with a center at (0,0) and a radius, r, is……..

222 ryx

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……..

222 )()( rkyhx

center (3,3)radius = 2

Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

Parabolas

© 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.

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

Why is the focus so important?

© Jill Britton, September 25, 2003

Parabola

• The Standard Form of a Parabola that opens to the right and has a vertex at (0,0) is……

axy 42

©1999 Addison Wesley Longman, Inc.

Parabola

• The Parabola that opens to the right and has a vertex at (0,0) has the following characteristics……

• a is the distance from the vertex of the parabola to the focus or directrix

• This makes the coordinates of the focus (a,0)• This makes the equation of the directrix x = -a• 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……

axy 42

© Shelly Walsh

Parabola

• The Parabola that opens to the left and has a vertex at (0,0) has the following characteristics……

• a is the distance from the vertex of the parabola to the focus or directrix

• This makes the coordinates of the focus(-a,0)• This makes the equation of the directrix x = a• 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……

ayx 42

©1999-2003 SparkNotes LLC, All Rights Reserved

Parabola

• The Parabola that opens up and has a vertex at (0,0) has the following characteristics……

• ‘a’ is the distance from the vertex of the parabola to the focus or directrix

• This makes the coordinates of the focus (0,a)• This makes the equation of the directrix y = -a• 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……

ayx 42

©1999 Addison Wesley Longman, Inc.

Parabola

• The Parabola that opens down and has a vertex at (0,0) has the following characteristics……

• ‘a’ is the distance from the vertex of the parabola to the focus or directrix

• This makes the coordinates of the focus (0,-a)• This makes the equation of the directrix y = a• 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……

)(4)( 2 hxpky

© 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

Parabola• The Standard Form of a Parabola that opens to the left

and has a vertex at (h,k) is……

)(4)( 2 hxpky

©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

Parabola

• The Standard Form of a Parabola that opens up and has a vertex at (h,k) is……

)(4)( 2 kyphx

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

•

•

Parabola

• The Standard Form of a Parabola that opens down and has a vertex at (h,k) is……

)(4)( 2 kyphx

Copyright ©1999-2004 Oswego City School District Regents Exam Prep Center

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.

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™.

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.

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.

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

Ellipse

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

12

2

2

2

b

y

a

x

Ellipse

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

12

2

2

2

a

y

b

x

Hyperbola

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.

© Jill Britton, September 25, 2003

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

Hyperbola

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

12

2

2

2

b

y

a

x

Hyperbola

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

12

2

2

2

b

x

a

y

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/ClassWebSite/2003%20student%20projects/27circlesandelipse.ppt

ResourcesFinney, 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/conics/conics.html

Mathews, Kevin. Great Buildings Online. Great Buildings. une 3, 2004 <http://www.GreatBuildings.com/buildings/Saint_Pauls_Cathedral.html

ResourcesMayoff, 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/math-topic.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.html

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