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Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0003.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
F1
F2
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Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
F2
F1
If P, = Q, and R are
point= s on a ellipse,
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Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
F2
F1
P
Q
R
If P, = Q, and R are
point= s on a ellipse,
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Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
F2
F1
P
Q
R
p1
p2
If P, = Q, and R are
point= s on a ellipse,
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Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
F2
F1
P
Q
R
p1
p2
If P, = Q, and R are
point= s on a ellipse,
q1
q2
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Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
F2
F1
P
Q
R
p1
p2
If P, = Q, and R are
point= s on a ellipse,
q1
q2
r2
r1
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Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
F2
F1
P
Q
R
p1
p2
If P, = Q, and R are
point= s on a ellipse,
then&= #13;
       p1 + p2
    =3D q= 1 + q2
 
q1
q2
r2
r1
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Ellipse= s
Given two fixed points (called = foci), an ellipse is the set of poi= nts whose sum of the distances to the foci is a constant.
F2
F1
P
Q
R
p1
p2
If P, = Q, and R are
point= s on a ellipse,
then&= #13;
       p1 + p2
    =3D q= 1 + q2
    =3D = r1  + r2
      =3D a constant<= /div>
q1
q2
r2
r1
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Ellipse= s
An ellipse has a center (h, k )= ;
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Ellipse= s
An ellipse has a center (h, k )= ;
(h, k)
(h, k)
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Ellipse= s
An ellipse has a center (h, k ); it has two
axes, the major (long) and the minor
(sh= ort) axes.
(h, k)
(h, k)
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Ellipse= s
An ellipse has a center (h, k ); it has two
axes, the major (l= ong) and the minor
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(h, k)
3D"Text
(h, k)
Major axis
Minor axis
Major axis
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Ellipse= s
An ellipse has a center (h, k ); it has two
axes, the major (l= ong) and the minor
(sh= ort) axes.
(h, k)
3D"Text
(h, k)
Major axis
Minor axis
Major axis
Minor axis
The horizontal axis is called the
x-radius, the vertical axis the y-radius.
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------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0016_image082.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhKwAXAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAYABgAd AA4AgAAAAP///wI5BHJoy+2r1GtBupgmvKgaf2jhNZbgE50qZFElu30tU83rJ3bt/ZJ9pznFXsDJ LCgzilzL5tDZVBQAADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0016_image083.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhgQAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwB8 AAYAgAAAAP///wIshI+py+0Po4xh2ouzxqGfDobiSJbmiabqyoZAC8fyTKvbjef6B+7+D3R4DgUA Ow== ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0016_image084.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDABHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMABAAG AEIAgAAAAP///wIwhGOHGa0OmpwQqhTXZTpj63FfB5bkOabiurWha6owyr52XOOzfNN77uvxdELg kFYAADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0016_image085.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDABzAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwAH AG4AgAAAAP///wJYhGMJgcl9XpTrMQdrplNx7V3YJkZKdZZmmGYoC2PtHK9y/d66zee96gsChy4i bef7KFVLkjP0RBpxU2nxevxhqVtrVtjVfqtiLvQ8QnfS7LUb5DXL43RDAQA7 ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0016_image086.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYgAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAIAAwBd AAYAgAAAAP///wImhI+py+0PWQhp2ouz3pzHr1xHR5bmiaYUyErq+7byTENijef4lBQAOx== ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0011.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipse= s
The gen= eral equation for ellipses is
Ax2 + By2 + Cx + Dy =3D E
where A= and B are the same sign but different numbers.
------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0017.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipse= s
The gen= eral equation for ellipses is
Ax2 + By2 + Cx + Dy =3D E =
where A= and B are the same sign but different numbers. Using completing the square, such equations may be transform to the standard form:
=
=
        = ;       
------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipse= s
(x – h)
(y – k)
a2
b2
The general equation for ellipses is
Ax2 + By2 + Cx + Dy =3D E
where A and B are the same sign but different numbers. Using completing the square, such equations may be transform to the standard form:
        = ;            &n= bsp;   +        = ;         =3D 1 
------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0018_image087.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhwwADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC/ AAIAgAAAAP///wIajI+py+3vgJy02osp3Lz7D4bikpVmOaZqUwAAOw== ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0019.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipse= s
(x – h)2
(y – k)2
a
b2
(h, k) is the center.
The general equation for ellipses is
Ax2 + By2 + Cx + Dy =3D E
where A and B are the same sign but different numbers. Using completing the square, such equations may be transform to the standard form:
        = ;            &n= bsp;   +        = ;         =3D 1 
------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0019_image088.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhwwADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC/ AAIAgAAAAP///wIajI+py+3vgJy02osp3Lz7D4bikpVmOaZqUwAAOw== ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0019_image089.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQAgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwBe ABsAgAAAAP///wJ6hI+poe0PU2Cx2jXp3TtrDh7eF5aSZ5Zoyhpre71wKs/PBJA2SOG74/vNgkKX rkg7DpVIE9GWaf6erah0ygxRr7CtNstNVsFhsZNcNvPQaXXM24ayjXEu/FQP3+l5fXbft+MFGCio Y1VY9kSYaCgy1yiXAxkpWdkXUAAAOw== ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0020.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
The gen= eral equation for ellipses is
Ax2 + By2 + Cx + Dy =3D E
where A= and B are the same sign but different numbers. Using completing the square, such equations may be transform to the standard form:
        = ;            &n= bsp;   +        = ;         =3D 1 =
Ellipse= s
(x – h)2
(y – k)2
a
b2
x-axis =3D a,   y-axis =3D b
(h, k) is the center.
------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0020_image090.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhwwADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAC/ AAIAgAAAAP///wIajI+py+3vgJy02osp3Lz7D4bikpVmOaZqUwAAOw== ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0020_image091.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhYQAgAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAwBe ABsAgAAAAP///wJ6hI+poe0PU2Cx2jXp3TtrDh7eF5aSZ5Zoyhpre71wKs/PBJA2SOG74/vNgkKX rkg7DpVIE9GWaf6erah0ygxRr7CtNstNVsFhsZNcNvPQaXXM24ayjXEu/FQP3+l5fXbft+MFGCio Y1VY9kSYaCgy1yiXAxkpWdkXUAAAOw== ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0020_image092.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhDAAcAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAQAAgAI ABYAgAAAAP///wIehB15GMsOlJMntUXrxVvPHHXPB1GWeUqpllmlGDkFADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0020_image093.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhEQAcAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAgAK ABYAgAAAAP///wIiTICJdprmXossrumqhRsrTXmSeFkAF5LnITKYi17gqMZZAQA7 ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0036.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipse= s
(x – h)2
(y – k)2
a
b2
x-axis =3D a,   y-axis =3D b
(h, k) is the center.
+
=3D 1
Example: Find the center,
major and minor axes. Draw and label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
4
22
+
=3D 1
------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0036_image094.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhtQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACy AAIAgAAAAP///wIYjI+py+3vgJy0Wguz3rz7Dy7XSE7hiTYFADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0036_image095.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh5QCIAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQDk AIYAgAAAAP///wL/jI+py+0Po5y02osv2Lz7D4biSJbmiabqyrap4cbyTNf2jQNwzvf+D8ztgsSi 8fgbIpfMpjOkfEqn1F60is1qUdet9+vtgsdkp7iMTgPP6rabxn7L59wA/Y5/2fP8/ifuF9gGKFhI RmiYuIWo2EjF6BjZBClZaURpmZm0p9lpxukZeoQpWupCapqqp8rKg9oKC/IaS7sxWxt7i9uqu6va 62sKHCw6TOxpfKyZrGzJ3Cz5DO0oPa1YbW2InS24ze3n/c0XLo5HXk53ji6nvu7W7q4GH482T38I ev+bry/M31/sH0BkAgcuK2jQGcKE0RYypObw4bWIErVRrNjtIkZwyBo3juvo0RzIkOlGkmRn8uS7 lCrlsWxZ7yVMfDMn1kxk7+YamToX8eyZJSdQVz+HPipqVIrQpDaWMoWD9CkTp1JjUK3a4irWFVq3 rvK6EmxYsS7JljUbE21atTTZjunq9k/UuDfg0uVg966OuXpn5L37l27guIPdFmZ7WG1itIvNNib7 WGxksJO9Vt56GWvmqpuldn76mWnopKONlh56Gmjqnqt1tr75umbsmbNh1m55W2Xuk7tJ9g7522OG 4cSLGz+OPLlyBAUAADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0036_image096.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhtQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACy AAIAgAAAAP///wIYjI+py+3vgJy0Wguz3rz7Dy7XSE7hiTYFADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0037.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipse= s
(x – h)2
(y – k)2
a
b2
x-axis =3D a,   y-axis =3D b
(h, k) is the center.
+
=3D 1
Example: Find the center,
major and minor axes. Draw and label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
4
22
+
=3D 1
Center: (3, -1)
------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0037_image097.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhtQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACy AAIAgAAAAP///wIYjI+py+3vgJy0Wguz3rz7Dy7XSE7hiTYFADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0037_image098.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh5QCIAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQDk AIYAgAAAAP///wL/jI+py+0Po5y02osv2Lz7D4biSJbmiabqyrap4cbyTNf2jQNwzvf+D8ztgsSi 8fgbIpfMpjOkfEqn1F60is1qUdet9+vtgsdkp7iMTgPP6rabxn7L59wA/Y5/2fP8/ifuF9gGKFhI RmiYuIWo2EjF6BjZBClZaURpmZm0p9lpxukZeoQpWupCapqqp8rKg9oKC/IaS7sxWxt7i9uqu6va 62sKHCw6TOxpfKyZrGzJ3Cz5DO0oPa1YbW2InS24ze3n/c0XLo5HXk53ji6nvu7W7q4GH482T38I ev+bry/M31/sH0BkAgcuK2jQGcKE0RYypObw4bWIErVRrNjtIkZwyBo3juvo0RzIkOlGkmRn8uS7 lCrlsWxZ7yVMfDMn1kxk7+YamToX8eyZJSdQVz+HPipqVIrQpDaWMoWD9CkTp1JjUK3a4irWFVq3 rvK6EmxYsS7JljUbE21atTTZjunq9k/UuDfg0uVg966OuXpn5L37l27guIPdFmZ7WG1itIvNNib7 WGxksJO9Vt56GWvmqpuldn76mWnopKONlh56Gmjqnqt1tr75umbsmbNh1m55W2Xuk7tJ9g7522OG 4cSLGz+OPLlyBAUAADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0037_image099.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhtQADAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAACy AAIAgAAAAP///wIYjI+py+3vgJy0Wguz3rz7Dy7XSE7hiTYFADs= ------=_NextPart_01C82DF8.A7350BF0 Content-Location: file:///C:/DEC9D913/6Ellipses_files/slide0038.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Ellipse= s
(x – h)2
(y – k)2
a
b2
x-axis =3D a,   y-axis =3D b
(h, k) is the center.
+
=3D 1
Example: Find the center,
major and minor axes. Draw and label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
4
22
+
=3D 1
Center: (3, -1)
Major axis is 4, its horizontal.
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Ellipse= s
(x – h)2
(y – k)2
a
b2
x-axis =3D a,   y-axis =3D b
(h, k) is the center.
+
=3D 1
Example: Find the center,
major and minor axes. Draw and label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
4
22
+
=3D 1
Center: (3, -1)
Major axis is 4, its horizontal.
Minor axis is 2<= span lang=3DEN-US style=3D'font-size:156%;text-shadow:auto;color:#FF9933;mso-fa= reast-language: ZH-CN'>, its vertical.
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Ellipse= s
(x – h)2
(y – k)2
a
b2
x-axis =3D a,   y-axis =3D b
(h, k) is the center.
+
=3D 1
Example: Find the center,
major and minor axes. Draw and label the top, bottom, right and left most points.
(x – 3)2
(y + 1)2
4
22
+
=3D 1
Center: (3, -1)
Major axis is 4, its horizontal.
Minor axis is 2<= span lang=3DEN-US style=3D'font-size:156%;text-shadow:auto;color:#FF9933;mso-fa= reast-language: ZH-CN'>, its vertical.
(3, -1)
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