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Hyperbolas
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Hyperbolas
Just as a= ll the other conic sections, hyperbolas are defined
by distan= ce relations.
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Hyperbolas
Given two= fixed points, called foci, a hyperbola is the set
of points= whose difference of the distances to the foci is
a constan= t.
Just as a= ll the other conic sections, hyperbolas are defined
by distan= ce relations.
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A
If A, B and C are points = on a hyperbola as shown
Hyperbolas
Given two= fixed points, called foci, a hyperbola is the set
of points= whose difference of the distances to the foci is
a constan= t.
B
C
Just as a= ll the other conic sections, hyperbolas are defined
by distan= ce relations.
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A
a2
a1
If A, B and C are points on a hyperbola as shown then   
a1 – a2 
Hyperbolas
Given two= fixed points, called foci, a hyperbola is the set
of points= whose difference of the distances to the foci is
a constan= t.
B
C
Just as a= ll the other conic sections, hyperbolas are defined
by distan= ce relations.
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A
a2
a1
b2
b1
If A, B and C are points on a hyperbola as shown then   
a1 – a2 =3D b1 – b2 
Hyperbolas
Given two= fixed points, called foci, a hyperbola is the set
of points= whose difference of the distances to the foci is
a constan= t.
B
C
Just as a= ll the other conic sections, hyperbolas are defined
by distan= ce relations.
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A
a2
a1
b2
b1
If A, B and C are points on a hyperbola as shown then   
a1 – a2 =3D b1 – b2 =3D c2  – c1 = =3D constant.
c1
c2
Hyperbolas
Given two= fixed points, called foci, a hyperbola is the set
of points= whose difference of the distances to the foci is
a constan= t.
B
C
Just as a= ll the other conic sections, hyperbolas are defined
by distan= ce relations.
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Hyperbolas
A hyperbola has a “center”, and two straight lines that
cradle the hyperbolas<= /span> <= /span>which are= <= /span>called asymptotes. =
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Hyperbolas
A hyperbola has a “center”, and two straight lines that
cradle the hyperbolas<= /span> <= /span>which are= <= /span>called asymptotes. =
There are two vertices= , one for each branch.
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Hyperbolas
A hyperbola has a “center”, and two straight lines that
cradle the hyperbolas<= /span> <= /span>which are= <= /span>called asymptotes. =
There are two vertices= , one for each branch. The asymptotes are the diagonals of a= box with the vertices of the hyperbola touching the box.
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Hyperbolas
A hyperbola has a “center”, and two straight lines that
cradle the hyperbolas<= /span> <= /span>which are= <= /span>called asymptotes. =
There are two vertices= , one for each branch. The asymptotes are the diagonals of a= box with the vertices of the hyperbola touching the box.
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Hyperbolas
A hyperbola has a “center”, and two straight lines that
cradle the hyperbolas<= /span> <= /span>which are= <= /span>called asymptotes. =
There are two vertices= , one for each branch. The asymptotes are the diagonals of a= box with the vertices of the hyperbola touching the box. The asymptotes are the diagonals of a box with the vertices of t= he hyperbola touching the box.
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a
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Hyperbolas
The center-box is defined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first.
a
b
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Hyperbolas
The center-box is defined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are= the asymptotes.
a
b
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Hyperbolas
The center-box is defined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are= the asymptotes. Label the vertices and trace the hyperbola= along the asympototes.
a
b
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Hyperbolas
The center-box is defined by the x-radius a, and y-radius b as shown. Hence, to graph a hyperbola, we find the center and the center-box first. Draw the diagonals of the box which are= the asymptotes. Label the vertices and trace the hyperbola= along the asympototes.
a
b
The location of the center, the x-radius a, and y-radius b may be obtained from the equation.
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Hyperbolas
The equations of hyper= bolas have the form
Ax2 + By2 + Cx + Dy =3D E
where A and B are oppo= site signs.
------=_NextPart_01C8FAF6.ABF5CA50 Content-Location: file:///C:/2A593C93/2Hyperbolas_files/slide0132.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
Hyperbolas
The equations of hyper= bolas have the form
Ax2 + By2 + Cx + Dy =3D E
where A and B are oppo= site signs. By completing the square, they may be transforme= d to the standard forms below.
------=_NextPart_01C8FAF6.ABF5CA50 Content-Location: file:///C:/2A593C93/2Hyperbolas_files/slide0133.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Ellipses
(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;
a2&#= 13;
b2&#= 13;
Hyperbolas
The equations of hyper= bolas have the form
Ax2 + By2 + Cx + Dy =3D E
where A and B are oppo= site signs. By completing the square, they may be transforme= d to the standard forms below.
– <= /span>
=3D 1 <= span lang=3DEN-US style=3D'font-size:133%;color:#0000CC;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
(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;
a2&#= 13;
b2&#= 13;
– <= /span>
=3D 1 <= span lang=3DEN-US style=3D'font-size:133%;color:#0000CC;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
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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;
a2&#= 13;
b2&#= 13;
(h, k) is the center.
Hyperbolas
The equations of hyper= bolas have the form
Ax2 + By2 + Cx + Dy =3D E
where A and B are oppo= site signs. By completing the square, they may be transforme= d to the standard forms below.
– <= /span>
=3D 1 <= span lang=3DEN-US style=3D'font-size:133%;color:#0000CC;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
(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;
a2&#= 13;
b2&#= 13;
– <= /span>
=3D 1 <= span lang=3DEN-US style=3D'font-size:133%;color:#0000CC;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
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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;
a2&#= 13;
b2&#= 13;
x-rad =3D= a,  y-rad =3D b=
(h, k) is the center.
Hyperbolas
The equations of hyper= bolas have the form
Ax2 + By2 + Cx + Dy =3D E
where A and B are oppo= site signs. By completing the square, they may be transforme= d to the standard forms below.
– <= /span>
=3D 1 <= span lang=3DEN-US style=3D'font-size:133%;color:#0000CC;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
(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;
a2&#= 13;
b2&#= 13;
– <= /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;
a2&#= 13;
b2&#= 13;
x-rad =3D= a,  y-rad =3D b=
(h, k) is the center.
Hyperbolas
The equations of hyper= bolas have the form
Ax2 + By2 + Cx + Dy =3D E
where A and B are oppo= site signs. By completing the square, they may be transforme= d to the standard forms below.
– <= /span>
=3D 1 <= span lang=3DEN-US style=3D'font-size:133%;color:#0000CC;mso-fareast-language:ZH= -CN; mso-special-format:lastCR;display:none'>
(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;