The Hyperbola
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Definition:
A hyperbola is the set of
points such that the difference of the distances from each point to two fixed
points (the foci) is constant. |
Click here for a demonstration
of the definition
Recall that for an ellipse the sum of the distances
from a point on the curve to the two foci remains constant, but note for a
hyperbola, it is the difference of the two distances that remains constant.
To make a Sketchpad model for
drawing a hyperbola, begin by opening the following Two-Pins
–and-a-String Construction for an ellipse.
Review the steps of construction and the reason this construction
works. It will not take much to change
this model so that it will draw a hyperbola.
Open the following construction to alter the ellipse sketch slightly
in order to draw a hyperbola.
The following information
should be given to the students through direct teaching
A hyperbola is the set of all points, P, in a plane
such that the difference from
P to two fixed points, called foci, is a constant.
The standard form of the hyperbola with center (h,k)
is
or 
They hyperbola is similar in some of its
characteristics to the ellipse, but very different in others.
The foci for the hyperbola are defined by the
relationship
The latus rectum relationship is the same for both the
hyperbola and the ellipse.
For each graph, list the
vertices, the foci, the latus rectum distance and the center. As you graph the following hyperbolas, watch
for similarities and differences between this curve
and the ellipse listed next
to it.
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Hyperbola |
Ellipse |
Answer |
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Answer the question and do
the activities that follow:
1.
What do you
notice about the relationship between the size of a and b
on the hyperbola that is different on the ellipse?
2.
What seems to
determine the direction of the hyperbola?
3.
What determines
the direction of the ellipse?
4.
Compare and
contrast as many characteristics of the two curves as you can.
5.
Write the
equations for each of the following hyperbolas and check them on the computer.
a.
hyperbola (h,k)
=(2,3)
a=4, b=9 lies in the x-direction
b.
hyperbola (h,k)
=(-1,-2)
a=9, b=4 lies in the y-direction
c.
hyperbola (h,k)
=(4,5)
a=4, c=16 lies in the y-direction