Constructing Hyperbolas

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Constructing Hyperbolas
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A hyperbola is created from the intersection of a
plane with a double cone.
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An example of a hyperbola occurring in nature is
shown in the picture. Two parallel glass plates
touching at the left end but with an opening of
about 5 mm between them at the right, are dipped
in beet juice. The juice rises by capillarity to
form a hyperbola. The reason that the juice
rises is due to the surface tension of the liquid
and this surface tension counteracts the force of
gravity.
The closer the glass is together the more the
surface tension is able to counteract gravity.
As you see, as the separation between the glass
increases the lower the beet juice falls.
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A hyperbola is defined by a group of points that
have a same difference of distance from two foci.
When you subtract the small line from the long
line for each ordered pair the remaining value is
the same.
Hyperbolas can be symmetrical around the x-axis
or the y-axis The one on the right is
symmetrical around the x-axis.
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The hyperbola is defined by two foci. A
transverse axis passes through both foci. A
conjugate axis is perpendicular to the transverse
axis through the centre (0,0). There are two
asymptotes that intersect each other through the
centre. They restrict the path of the hyperbola.
The form of this horizontal hyperbola is
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Vertices (a,0) and (-a,0)
c2 a2 b2
Foci (c,0) and (-c,0)
Vertices (0,a) and (0,-a)
c2 a2 b2
Foci (0,c) and (0,-c)
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a2 16 a 4
b2 9 b 3
Vertices (4,0) and (-4,0)
c2 a2 b2 c2 16 9 c2 25 c 5
Foci (5,0) and (-5,0)
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Vertices (0,4) and (0,-4)
Foci (0,5) and (0,-5)
a2 16 a 4
b2 9 b 3
c2 a2 b2 c2 16 9 c2 25 c 5
Asymptotes
Domain R
Range -?,-4 ? 4, ?
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Test (0,0)
0 lt 1
True
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le fin
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final
The end
finito
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