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Click to edit Master title style
Clic= k to edit Master text styles
Sec= ond level
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‹date/time›&= #13;
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------=_NextPart_01C9B783.7C531BF0 Content-Location: file:///C:/2A593C93/38EllipsesandHyperbolas_files/master06.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9B783.7C531BF0 Content-Location: file:///C:/2A593C93/38EllipsesandHyperbolas_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAADvAAAAAUAHAAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwC7wLpAwMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAMAAAAQAkhAPAAAAAAAAAA 7wLpA/////8IAAAA+gIAAAAAAAAAAAAABAAAAC0BAgAEAAAALQEAAAQAAAAnAf//HAAAAPsC1P8A AAAAAACQAQAAAAAAQAAAQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEAAAALQEDAAQA AAAuARgABAAAAAIBAQAFAAAACQIAAAACEwAAADIKNgCaAQgAAABFbGxpcHNlcx0ACgAKAAoAGQAW ABgAFgAEAAAALgEAABwAAAD7AhAABwAAAAAAvAIAAAAAAQICIlN5c3RlbQAAAAAAABgAAAAwnhMA AQAAAOQEAAAAAAAABAAAAC0BBAAEAAAA8AEDAAMAAAAAAA== ------=_NextPart_01C9B783.7C531BF0 Content-Location: file:///C:/2A593C93/38EllipsesandHyperbolas_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C9B783.7C531BF0 Content-Location: file:///C:/2A593C93/38EllipsesandHyperbolas_files/slide0133.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Conic Sections
Ellipses
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Ellipses
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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F2&= #13;
F1&= #13;
Ellipses
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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F2&= #13;
F1&= #13;
P
Q
R
If P, Q, and R are any
points on a ellipse,
Ellipses
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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F2&= #13;
F1&= #13;
P
Q
R
p1&= #13;
p2&= #13;
If P, Q, and R are any
points on a ellipse, then
p1 + p2
Ellipses
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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F2&= #13;
F1&= #13;
P
Q
R
p1&= #13;
p2&= #13;
If P, Q, and R are any
points on a ellipse, then
p1 + p2
=3D q1 + q2
q1
q2
Ellipses
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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F2&= #13;
F1&= #13;
P
Q
R
p1&= #13;
p2&= #13;
If P, Q, and R are any
points on a ellipse, then
p1 + p2
=3D q1 + q2
=3D r1  + r2
q1
q2
r2
r1
Ellipses
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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F2&= #13;
F1&= #13;
P
Q
R
p1&= #13;
p2&= #13;
If P, Q, and R are any
points on a ellipse, then
p1 + p2
=3D q1 + q2
=3D r1  + r2
=3D a constant
q1
q2
r2
r1
Ellipses
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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F2&= #13;
F1&= #13;
P
Q
R
p1&= #13;
p2&= #13;
If P, Q, and R are any
points on a ellipse, then
p1 + p2
=3D q1 + q2
=3D r1  + r2
=3D a constant
q1
q2
r2
r1
Ellipses
An ellipse has a center (h, k );
(h, k)
(h, k)
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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F2&= #13;
F1&= #13;
P
Q
R
p1&= #13;
p2&= #13;
If P, Q, and R are any
points on a ellipse, then
p1 + p2
=3D q1 + q2
=3D r1  + r2
=3D a constant
q1
q2
r2
r1
Ellipses
An ellipse has a center (h, k ); it has t= wo axes, the major (long)
(h, k)
(h, k)
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
Major axis
Major axis
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F2&= #13;
F1&= #13;
P
Q
R
p1&= #13;
p2&= #13;
If P, Q, and R are any
points on a ellipse, then
p1 + p2
=3D q1 + q2
=3D r1  + r2
=3D a constant
q1
q2
r2
r1
Ellipses
An ellipse has a center (h, k ); it has t= wo axes, the major (long) and the minor (short) axes.
(h, k)
Major axis
Minor axis
(h, k)
Major axis
Minor axis
Given two fixed points (called = foci), an ellipse is the set of = 3;
points whose sum of the distanc= es to the foci is a constant.
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These axes correspond to the important radii of the ellipse.
Ellipses
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The= se axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius
Ellipses
x-radius
x-radius
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y-radius
The= se axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius.
Ellipses
x-radius
x-radius
y-radius
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The= se axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius.
Ellipses
x-radius
The general equation for ellips= es is
Ax2 + By2 + Cx + Dy =3D E =
where A and B are the same sign= but different numbers. = ;
x-radius
y-radius
y-radius
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The= se axes correspond to the important radii of the ellipse. From the center, the horizontal length is called the x-radius and the vertical length the y-radius.
Ellipses
x-radius
The general equation for ellips= es is
Ax2 + By2 + Cx + Dy =3D E =
where A and B are the same sign= but different numbers. Using completing the square<= /i>, such equations may be = transform to the standard form of ellipses below.
x-radius
y-radius
y-radius
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(x – h)2
(y – k)2
a2
b2
Ellipses
+
=3D 1
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
Ellipses
+
=3D 1
This has to be 1.
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
(h, k) is the center.
Ellipses
+
=3D 1
This has to be 1.
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
x-radius =3D a   
(h, k) is the center.
Ellipses
+
=3D 1
This has to be 1.
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
x-radius =3D a    y-radius =3D b=
(h, k) is the center.
Ellipses
+
=3D 1
This has to be 1.
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
x-radius =3D a    y-radius =3D b=
(h, k) is the center.
Ellipses
+
=3D 1
This has to be 1.
Example A. Find the center, major and minor axes. Draw and <= /span>label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
42
22
+
=3D 1
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
x-radius =3D a    y-radius =3D b=
(h, k) is the center.
Ellipses
+
=3D 1
This has to be 1.
Example A. Find the center, major and minor axes. Draw and <= /span>label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
42
22
+
=3D 1
The center is (3, –1).
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
x-radius =3D a    y-radius =3D b=
(h, k) is the center.
Ellipses
+
=3D 1
This has to be 1.
Example A. Find the center, major and minor axes. Draw and <= /span>label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
42
22
+
=3D 1
The center is (3, –1).
The x-radius is 4.
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
x-radius =3D a    y-radius =3D b=
(h, k) is the center.
Ellipses
+
=3D 1
This has to be 1.
Example A. Find the center, major and minor axes. Draw and <= /span>label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
42
22
+
=3D 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2. <= /div>
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
x-radius =3D a    y-radius =3D b=
(h, k) is the center.
Ellipses
+
=3D 1
This has to be 1.
(3, -1)
Example A. Find the center, major and minor axes. Draw and <= /span>label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
42
22
+
=3D 1
The center is (3, –1).
The x-radius is 4.
The y-radius is 2. <= /div>
The Standard Form <= /div>
(of Ellipses)
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(x – h)2
(y – k)2
a2
b2
x-radius =3D a    y-radius =3D b=
(h, k) is the center.