BY:-NEERAJ CHAURASIA PGT (MATHS) KENDRIYA VIDYALAYA DIBRUGARH (ASSAM)

What’s in a Parabola A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane. A parabola is the set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane. Copyright © , Math Academy Online™ / Platonic Realms™.

Parabola The Standard Form of a Parabola that opens to the right and has a vertex at (0,0) is…… 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…… The Parabola that opens to the right and has a vertex at (0,0) has the following characteristics…… Y 2 = 4pX Y 2 = 4pX p is the distance from the vertex of the parabola to the focus or directrix 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 coordinates of the focus (p,0) This makes the equation of the directrix x = -p This makes the equation of the directrix x = -p The makes the axis of symmetry the x-axis (y = 0) 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…… 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…… 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 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 coordinates of the focus(-p,0) This makes the equation of the directrix x = p This makes the equation of the directrix x = p The makes the axis of symmetry the x-axis (y = 0) 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…… The Standard Form of a Parabola that opens up and has a vertex at (0,0) is…… © SparkNotes LLC, All Rights Reserved

Parabola The Parabola that opens up and has a vertex at (0,0) has the following characteristics…… 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 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 coordinates of the focus (0,p) This makes the equation of the directrix y = -p This makes the equation of the directrix y = -p This makes the axis of symmetry the y-axis (x = 0) 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…… 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…… 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 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 coordinates of the focus (0,-p) This makes the equation of the directrix y = p This makes the equation of the directrix y = p This makes the axis of symmetry the y-axis (x = 0) 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…… 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…….. 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 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 coordinates of the focus (h+p, k) This makes the equation of the directrix x = h – p This makes the equation of the directrix x = h – p This makes the axis of symmetry 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…… 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…… 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 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 coordinates of the focus (h – p, k) This makes the equation of the directrix x = h + p This makes the equation of the directrix x = h + p The makes the axis of symmetry The makes the axis of symmetry

Parabola The Standard Form of a Parabola that opens up and has a vertex at (h,k) is…… 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…… 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 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 coordinates of the focus (h, k + p) This makes the equation of the directrix y = k – p This makes the equation of the directrix y = k – p The makes the axis of symmetry The makes the axis of symmetry

Parabola The Standard Form of a Parabola that opens down and has a vertex at (h,k) is…… 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…… 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 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 coordinates of the focus (h, k - p) This makes the equation of the directrix y = k + p This makes the equation of the directrix y = k + p This makes the axis of symmetry 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.) 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. 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. 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 General Rules x and y are both squared x and y are both squared Equation always equals(=) 1 Equation always equals(=) 1 Equation is always plus(+) Equation is always plus(+) a 2 is always the biggest denominator a 2 is always the biggest denominator c 2 = a 2 – b 2 c 2 = a 2 – b 2 c is the distance from the center to each foci on the major axis 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. The center is in the middle of the 2 vertices, the 2 covertices, and the 2 foci.

Ellipse General Rules General Rules a is the distance from the center to each vertex on the major axis 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) b is the distance from the center to each vertex on the minor axis (co-vertices) Major axis has a length of 2a Major axis has a length of 2a Minor axis has a length of 2b 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) 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…… 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 axis has the following characteristics…… The ellipse with a center at (0,0) and a horizontal axis has the following characteristics…… Vertices ( a,0) Vertices ( a,0) Co-Vertices (0, b) Co-Vertices (0, b) Foci ( c,0) Foci ( c,0) © Cabalbag, Porter, Chadwick, and Liefting

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

Ellipse The ellipse with a center at (0,0) and a vertical axis has the following characteristics…… The ellipse with a center at (0,0) and a vertical axis has the following characteristics…… Vertices (0, a) Vertices (0, a) Co-Vertices ( b,0) Co-Vertices ( b,0) Foci (0, c) Foci (0, c) © Cabalbag, Porter, Chadwick, and Liefting

Ellipse The standard form of the ellipse with a center at (h,k) and a horizontal axis is…… 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…… The ellipse with a center at (h,k) and a horizontal axis has the following characteristics…… Vertices (h a, k) Vertices (h a, k) Co-Vertices (h, k b) Co-Vertices (h, k b) Foci (h c, k) Foci (h c, k) ©Sellers, James

Ellipse The standard form of the ellipse with a center at (h,k) and a vertical axis is…… 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…… The ellipse with a center at (h,k) and a vertical axis has the following characteristics…… Vertices (h, k a) Vertices (h, k a) Co-Vertices (h b, k) Co-Vertices (h b, k) Foci (h, k c) Foci (h, k c) © Joan Bookbinder

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. 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. 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 General Rules x and y are both squared x and y are both squared Equation always equals(=) 1 Equation always equals(=) 1 Equation is always minus(-) Equation is always minus(-) a 2 is always the first denominator a 2 is always the first denominator c 2 = a 2 + b 2 c 2 = a 2 + b 2 c is the distance from the center to each foci on the major axis 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 a is the distance from the center to each vertex on the major axis

Hyperbola General Rules 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). 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 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 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 x 2 is first then the hyperbola is horizontal If y 2 is first then the hyperbola is vertical. If y 2 is first then the hyperbola is vertical.

Hyperbola General Rules General Rules The center is in the middle of the 2 vertices and the 2 foci. 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 used to draw the rectangles that form the asymptotes. The vertices and the covertices are the midpoints of the rectangle 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 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…… 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…… The Hyperbola with a center at (0,0) and a horizontal axis has the following characteristics…… Vertices ( a,0) Vertices ( a,0) Foci ( c,0) Foci ( c,0) Asymptotes: Asymptotes:

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

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

Hyperbola The standard form of the Hyperbola with a center at (h,k) and a horizontal axis is…… 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…… The Hyperbola with a center at (h,k) and a horizontal axis has the following characteristics…… Vertices (h a, k) Vertices (h a, k) Foci (h c, k ) Foci (h c, k ) Asymptotes: Asymptotes:

Hyperbola The standard form of the Hyperbola with a center at (h,k) and a vertical axis is…… 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…… The Hyperbola with a center at (h,k) and a vertical axis has the following characteristics…… Vertices (h, k a) Vertices (h, k a) Foci (h, k c) Foci (h, k c) Asymptotes: Asymptotes: ©Sellers, James

Rotating the Coordinate Axis © James Wilson

Equations for Rotating the Coordinate Axes