MATHEMATICS PROJECT

DONE, HARI M XI B K.V.PATTOM

CONIC SECTIONS

The four basic conic sections are all created by cutting a double cone at different angles.

CIRCLES

©National Science Foundation

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

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

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.

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

pxy 42

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)

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

pxy 42

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)

The Standard Form of a Parabola that opens up and has a vertex at (0,0) is……

pyx 42

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)

The Standard Form of a Parabola that opens down and has a vertex at (0,0) is……

pyx 42

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)

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

)(4)( 2 hxpky

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

a

by

2

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

)(4)( 2 hxpky

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

by

2

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

)(4)( 2 kyphx

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 a

bx

2

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

)(4)( 2 kyphx

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

a

bx

2

© 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.

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 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.

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.

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.

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)

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

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)

1916

22

yx

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

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

yx

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

1)()(

2

2

2

2

b

ky

a

hx

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)

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

1)()(

2

2

2

2

a

ky

b

hx

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)

At St. Paul’s Cathedral in London, one can find an interesting example of an ellipse. In the Whispering gallery, a person can stand on the foci and whisper to another person on the other foci and they will be heard. The Whispering Gallery is located where at 99 feet above the ground.

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

The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.

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.

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

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.

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

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

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

Vertices ( a,0) Foci ( c,0)

Asymptotes: xa

by

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

The Hyperbola with a center at (0,0) and a vertical axis has the following characteristics……

Vertices (0, a) Foci ( 0, c)

Asymptotes: xb

ay

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

1)()(

2

2

2

2

b

ky

a

hx

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:

)( hxa

bky

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

1)()(

2

2

2

2

b

hx

a

ky

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: )( hxb

aky