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…….. center (0,0) radius = 2 Copyright © 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…….. center (3,3) radius = 2 Copyright © 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 © , 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…… ©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…… © 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…… So… 4p = the coefficient of y

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…… ©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…… © 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

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

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

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

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

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

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

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 © , 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.

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 (Formula for finding the foci) –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.

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.Artifice, Inc.

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……

Ellipse The ellipse with a center at (0,0) and a horizontal major axis has the following characteristics…… Vertices ( a,0) Co-Vertices (0, b) Foci ( c,0)

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

Ellipse The ellipse with a center at (0,0) and a vertical axis has the following characteristics…… Vertices (0, 9) Co-Vertices ( 3,0) Foci (0, ) C 2 = a 2 – b 2

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

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……

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) What is the center? What is the equation?

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) Center = (3,-1) Equation: Now find the foci

Ellipse The ellipse with a center at (h,k) and a vertical axis has the following characteristics… Vertices (3, 7) (3,-9) Co-Vertices (-3, -1) (9,-1) Foci: Center = (3,-1) Equation:

Eccentricity The closer “e” is to 0, the more circular an ellipse is. Find e:

Eccentricity The closer “e” is to 0, the more circular an ellipse is. Find e:

Identifying a conic Do Now: Find the equation of the circle in standard form, the radius and the center:

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 © , 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(-) –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. The axis through these points is the “conjugate axis” –Major axis (Transverse) has a length of 2a –Eccentricity (e): e = c/a –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……

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

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

Hyperbola Example of an hyperbola with a center at (0,0) and a vertical axis is…… Graph all vertices and make a rectangle:

Hyperbola The Hyperbola with a center at (0,0) and a vertical axis has the following characteristics…… Vertices (0, 8) Foci Asymptotes:

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

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:

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

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

Rotating the Coordinate Axis

Equations for Rotating the Coordinate Axes