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.. 2

2 x y 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..

2 2 ( x h) ( y k ) r center (3,3) radius = 2 Copyright 1999-2004 Oswego City School District Regents Exam Prep Center 2 Parabolas

Art Mayoff Long Island Fountain Company Whats 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 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 2 ( y k ) 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 2a Parabola The Standard Form of a Parabola that opens to the left

and has a vertex at (h,k) is 2 ( y k ) 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 2a Parabola The Standard Form of a Parabola that opens up and has a vertex at (h,k) is 2

( x h) 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

b x 2a 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 b x 2a

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

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 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 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 2 2 x y

2 2 a b 1 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) x2 y2 1 16 9 Cabalbag, Porter, Chadwick, and Liefting Ellipse The standard form of the ellipse with a center at (0,0)

and a vertical axis is 2 2 x y 1 2 2 b

a 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) x2 y2 1 9 81

Cabalbag, Porter, Chadwick, and Liefting Ellipse The standard form of the ellipse with a center at (h,k) and a horizontal axis is 2 2 ( x h) ( y k ) 1

2 2 a b 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 2 2 ( x h) ( y k ) 1 2 2

b a 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

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 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. 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 2 2 x y 1 2

2 a b Hyperbola The Hyperbola with a center at (0,0) and a horizontal axis has the following characteristics Vertices ( a,0)

Foci (c,0) Asymptotes: b y x a Hyperbola The standard form of the Hyperbola with a center at

(0,0) and a vertical axis is 2 2 y x 1 2 2 a

b Hyperbola The Hyperbola with a center at (0,0) and a vertical axis has the following characteristics Vertices (0, a) Foci ( 0, c)

Asymptotes: a y x b Hyperbola The standard form of the Hyperbola with a center at (h,k) and a horizontal axis is

2 2 ( x h) ( y k ) 1 2 2 a b 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:

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

( y k ) ( x h) 1 2 2 a b 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: a y k ( x h)

b Sellers, James Rotating the Coordinate Axis 2 2 Ax Bxy Cy Dx Ey F 0 James Wilson Equations for Rotating the Coordinate Axes

x x' cos y ' sin y x' sin y ' cos A C cot 2 B or B tan 2 A C

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). 2001. June3, 2004

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

June 3, 2004 http://mathworld.wolfram.com/ConicSection.html>. Mueller, William. Modeling Periodicity . June 3, 2004 . PRIME Articles. Platomic Realms. June 3, 2004 . Resources Quadratics. Spark Notes from Barnes and Noble. June 3, 2004

Regents Exam Prep. June, 3, 2004 . Seek One Web Services, Long Island Fountain Company. . 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