parabola of analytic geometry

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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

r y x

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

)()( r k yh x

center (3,3)

radius = 2




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 rightand has a vertex at (0,0) is……

px y 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……

• p is the distance from the vertex of the parabola to thefocus 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……

px y 42

© Shelly Walsh



Parabola

• The Parabola that opens to the left and has a vertex at

(0,0) has the following characteristics……


• 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 andhas a vertex at (0,0) is……

py x 4

2

©1999-2003 SparkNotes LLC, All Rights Reserved



Parabola

• The Parabola that opens up and has a vertex at (0,0)

has the following characteristics……


• 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……

py x 42

©1999 Addison Wesley Longman, Inc.



Parabola

• The Parabola that opens down and has a vertex at (0,0)



• 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……

)(4)( 2 h x pk y

© Shel ly Walsh



Parabola

• The Parabola that opens to the right and has a vertex at

(h,k) has the following characteristics……..


• 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

a

b y

2



Parabola

• The Standard Form of a Parabola that opens to the leftand has a vertex at (h,k) is……

)(4)(2

h x pk y

©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

a

b y

2



Parabola

• The Standard Form of a Parabola that opens up and has

a vertex at (h,k) is……

)(4)( 2 k y ph x




Parabola

• The Parabola that opens up and has a vertex at (h,k) has

the following characteristics……


• 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 symmetrya

b x

2



Parabola

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

)(4)(2

k y ph x




Parabola

• The Parabola that opens down and has a vertex at (h,k) has thefollowing 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 symmetrya

b x

2

Elli



Ellipse


•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 acousticalphenomenon. He situated his desk at a focal point of the elliptical ceiling, easily

eavesdropping on the private conversations of other House members located near theother 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 isused 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 isused in lithotripsy, a medical procedure for treating kidney stones. The patient is placed ina elliptical tank of water, with the kidney stone

at one focus. High-energy shock wavesgenerated at the other focus are concentratedon the stone, pulverizing it.

Wh th f i f th lli



Why are the foci of the ellipse

important?

• St. Paul's Cathedral in London. If a personwhispers 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.

http://www.artifice.com/about_artifice.html

http://www.artifice.com/about_artifice.html



Ellipse

• General Rules – x and y are both squared

– Equation always equals(=) 1

– Equation is always plus(+) – 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.



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 eachvertex 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

1, 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……

12

2

2

2

b

y

a

x



Ellipse

• The ellipse with a center at (0,0) and a horizontal axis


• Vertices ( a,0)

• Co-Vertices (0, b)

• Foci ( c,0)

1916

22

y x

© Cabalbag, Porter, Chadwick, and Liefting



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



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)

1819

22

y x

© Cabalbag, Porter, Chadwick, and Liefting



Ellipse

• The standard form of the ellipse with a center at (h,k)

and a horizontal axis is……

1)()(

2

2

2

2

b

k y

a

h x



Ellipse

• The ellipse with a center at (h,k) and a horizontal axishas 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……

1

)()(

2

2

2

2

a

k y

b

h x



Ellipse

• The ellipse with a center at (h,k) and a vertical axishas the following characteristics……

• Vertices (h, k a)

• Co-Vertices (h b , k)

• Foci (h, k c)

© Joan Bookbinder 1998 -2000



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.




What is a Hyperbola?

• The set of all points in the plane, thedifference 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 onthe 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.




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



Hyperbola

• General Rules – b is the distance from the center to each

midpoint of the rectangle used to draw theasymptotes. This distance runs

perpendicular to the distance (a). – Major axis has a length of 2a

– Eccentricity(e): e = c/a (The closer e gets to1, 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.



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 drawthe rectangles that form the asymptotes.

– The vertices and the covertices are the midpoints

of the rectangle

– The covertices are not labeled on the hyperbolabecause 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……

12

2

2

2

b

y

a

x



Hyperbola

• The Hyperbola with a center at (0,0) and a horizontal

axis has the following characteristics……

• Vertices ( a,0)

• Foci ( c,0)

• Asymptotes: x

a

b y



Hyperbola


(0,0) and a vertical axis is……

12

2

2

2

b

x

a

y



Hyperbola

• The Hyperbola with a center at (0,0) and a vertical


• Vertices (0, a)

• Foci ( 0, c)

• Asymptotes:

xb

a y



Hyperbola


(h,k) and a horizontal axis is……

1

)()(2

2

2

2

b

k y

a

h x



Hyperbola

• The Hyperbola with a center at (h,k) and a horizontalaxis has the following characteristics……

• Vertices (h a, k)

• Foci (h c, k )

• Asymptotes:

)( h x

a

bk y



Hyperbola


(h,k) and a vertical axis is……

1

)()(2

2

2

2

b

h x

a

k y



Hyperbola

• The Hyperbola with a center at (h,k) and a vertical


• Vertices (h, k a)• Foci (h, k c)

• Asymptotes:

)( h xb

a

k y

©Sellers, James



Rotating the Coordinate Axis

022 F Ey DxCy Bxy Ax

© James Wilson

E ti f R t ti th C di t



Equations for Rotating the Coordinate

Axes

sin'cos' y x x

cos'sin' y x y

B

C A 2cot C A

B

2tanor



Resources

Bookbinder, John. Unit 8: Conic Sections (College AlgebraOnline). 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 andLiefting. Nick. Graphing Conic Sections (Microsoft Power PointPresentation 1997). 2001. June3, 2004

<http://www.granite.k12.ut.us/Hunter_High/StaffPages/Olsen_P/ClassWebSite/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, 2004http://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>.

http://mathworld.wolfram.com/ConicSection.html



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.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 WebResource. http://mathworld.wolfram.com/ConicSection.html

Wilson, James W. CURVE BUILDING. An Exploration with

Algebraic Relations University of Georgia.

June 3, 2004http://jwilson.coe.uga.edu/Texts.Folder/cb/curve.building.html

http://mathworld.wolfram.com/

http://mathworld.wolfram.com/

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parabola of analytic geometry