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------=_NextPart_01C8FAF6.2950D040 Content-Location: file:///C:/DEC9D913/6Ellipses_files/master06.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C8FAF6.2950D040 Content-Location: file:///C:/DEC9D913/6Ellipses_files/preview.wmf Content-Transfer-Encoding: base64 Content-Type: image/x-wmf AQAJAAADxQAAAAYAHAAAAAAABAAAAAMBCAAFAAAACwIAAAAABQAAAAwCeQChAAMAAAAeAAcAAAD8 AgAA////AAAABAAAAC0BAAAIAAAA+gIFAAAAAAD///8ABAAAAC0BAQAOAAAAJAMFAP///////3gA oAB4AKAA////////CAAAAPoCAAAAAAAAAAAAAAQAAAAtAQIABwAAAPwCAAD///8AAAAEAAAALQED AAQAAAAnAf//HAAAAPsC+f8AAAAAAACQAQAAAAAAQAAiQXJpYWwAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAEAAAALQEEAAQAAAAuARgABAAAAAIBAQAFAAAACQIAAAACEwAAADIKCABCAAgAAABF bGxpcHNlcwUAAgABAAIABAAEAAMABAAEAAAALgEAABwAAAD7AhAABwAAAAAAvAIAAAAAAQICIlN5 c3RlbQAAAAAAAAAAAAAYAAAAAgAAAKj9EwLkBAAABAAAAC0BBQAEAAAA8AEEAAMAAAAAAA== ------=_NextPart_01C8FAF6.2950D040 Content-Location: file:///C:/DEC9D913/6Ellipses_files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C8FAF6.2950D040 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0010.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipses<= /div>
------=_NextPart_01C8FAF6.2950D040 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0051.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipses<= /div>
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
------=_NextPart_01C8FAF6.2950D040 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0052.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
F2
F1
Ellipses<= /div>
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
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F2
F1
P
Q
R
If P, Q, and R are any=
points on a ellipse, <= /span>
Ellipses<= /div>
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
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F2
F1
P
Q
R
p1
p2
If P, Q, and R are any=
points on a ellipse, then
p1 = + p2
Ellipses<= /div>
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
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F2
F1
P
Q
R
p1
p2
If P, Q, and R are any=
points on a ellipse, then
p1 = + p2
=3D q= 1 + q2
q1
q2
Ellipses<= /div>
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
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F2
F1
P
Q
R
p1
p2
If P, Q, and R are any=
points on a ellipse, then
p1 = + p2
=3D q= 1 + q2
=3D = r1  + r2
q1
q2
r2
r1
Ellipses<= /div>
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
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F2
F1
P
Q
R
p1
p2
If P, Q, and R are any=
points on a ellipse, then
p1 = + p2
=3D q= 1 + q2
=3D = r1  + r2
=3D a constant<= /div>
q1
q2
r2
r1
Ellipses<= /div>
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
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F2
F1
P
Q
R
p1
p2
If P, Q, and R are any=
points on a ellipse, then
p1 = + p2
=3D q= 1 + q2
=3D = r1  + r2
=3D a constant<= /div>
q1
q2
r2
r1
Ellipses<= /div>
An ellipse has a center (h, k );
(h, k)
(h, k)
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
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F2
F1
P
Q
R
p1
p2
If P, Q, and R are any=
points on a ellipse, then
p1 = + p2
=3D q= 1 + q2
=3D = r1  + r2
=3D a constant<= /div>
q1
q2
r2
r1
Ellipses<= /div>
An ellipse has a cente= r (h, k ); it has two axes, the major (long) <= /div>
(h, k)
(h, k)
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
Major a= xis
Major a= xis
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F2
F1
P
Q
R
p1
p2
If P, Q, and R are any=
points on a ellipse, then
p1 = + p2
=3D q= 1 + q2
=3D = r1  + r2
=3D a constant<= /div>
q1
q2
r2
r1
Ellipses<= /div>
An ellipse has a cente= r (h, k ); it has two axes, the major (long) and the minor (= short) axes.
(h, k)
Major a= xis
Minor a= xis
(h, k)
Major a= xis
Minor a= xis
Given two= fixed points (called foci), an ellipse is the s= et of
points wh= ose sum of the distances to the foci is a constant.
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These axes correspond to the important radii of the ellipse.
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These axes correspond to the important radii of the ellipse. From the = center, the horizontal length is called the x-radius
Ellipses<= /div>
x-radius<= /span>
x-radius<= /span>
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y-radius<= /span>
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Ellipses<= /div>
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x-radius<= /span>
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These axes correspond to the important radii of the ellipse. From the = center, the horizontal length is called the x-radius and the v= ertical length the y-rad= ius. =
Ellipses<= /div>
x-radius<= /span>
The gener= al equation for ellipses is
Ax2 + By2 + Cx + Dy =3D E
where A a= nd B are the same sign but different numbers.
x-radius<= /span>
y-radius<= /span>
y-radius<= /span>
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These axes correspond to the important radii of the ellipse. From the = center, the horizontal length is called the x-radius and the v= ertical length the y-rad= ius. =
Ellipses<= /div>
x-radius<= /span>
The gener= al equation for ellipses is
Ax2 + By2 + Cx + Dy =3D E
where A a= nd B are the same sign but different numbers. Using comple= ting the square, such equations may be transform to the standa= rd form of ellips= es below.<= /span>
x-radius<= /span>
y-radius<= /span>
y-radius<= /span>
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
Ellipses<= /div>
+<= /div>
=3D 1
The Stand= ard Form
(of Ellip= ses)
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
Ellipses<= /div>
+<= /div>
=3D 1
This has = to be 1.
The Stand= ard Form
(of Ellip= ses)
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(h, k) is the c= enter.
Ellipses<= /div>
+<= /div>
=3D 1
This has = to be 1.
The Stand= ard Form
(of Ellip= ses)
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
x-radius = =3D a   
(h, k) is the c= enter.
Ellipses<= /div>
+<= /div>
=3D 1
This has = to be 1.
The Stand= ard Form
(of Ellip= ses)
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
x-radius = =3D a    y-radius =3D b=
(h, k) is the c= enter.
Ellipses<= /div>
+<= /div>
=3D 1
This has = to be 1.
The Stand= ard Form
(of Ellip= ses)
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
x-radius = =3D a    y-radius =3D b=
(h, k) is the c= enter.
Ellipses<= /div>
+<= /div>
=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)
(y + 1)2
42
22
+
=3D 1
The Stand= ard Form
(of Ellip= ses)
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
x-radius = =3D a    y-radius =3D b=
(h, k) is the c= enter.
Ellipses<= /div>
+<= /div>
=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)
(y + 1)2
42
22
+
=3D 1
The center is (3, –1).
The Stand= ard Form
(of Ellip= ses)
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
x-radius = =3D a    y-radius =3D b=
(h, k) is the c= enter.
Ellipses<= /div>
+<= /div>
=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)
(y + 1)2
42
22
+
=3D 1
The cent= er is (3, –1).
The x-ra= dius is 4.
The Stand= ard Form
(of Ellip= ses)
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(x –= ; h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
(y –= ; k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
a<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
b<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2&#= 13;
x-radius = =3D a    y-radius =3D b=
(h, k) is the c= enter.
Ellipses<= /div>
+<= /div>
=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)
(y + 1)2
42
22
+
=3D 1
The cent= er is (3, –1).
The x-ra= dius is 4.
The y-ra= dius is 2.
The Stand= ard Form
(of Ellip= ses)
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