Download Hyperbolas

Math 454
Tuesday 9/12/2000
Reminder that if you have the NTE books (study tips, solutions to practice exam, approaches to
problem solving) you will need to pass them on to others in the class in approx 2 weeks. You
may keep the copy of the NTE practice test you receive today.
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Reminder of solution of systems of n linear equations in n unknowns on calculator # 17
Section 13.1 p. 511
Questions on 13.3 homework?
Section 13.4 Hyperbolas Def: A hyperbola is a set of points such that the absolute value of the difference of the
distances from each point to two given points called the foci is a constant.
Draw illustration: (Include foci)
Terms: Fundamental
Rectangle,
Asymptotes,
Principal Axis (line
connecting the foci);
Vertices (points of
intersection of the
hyperbola with the
principal axis), Center
(midpoint of the
segment connecting
the vertices),
transverse axis
(segment whose
endpoints are the
vertices, length of
transverse axis is one
dimension of the
fundamental
rectangle); conjugate
axis (segment
perpendicular to the
conjugate axis, length
is the other dimension
of the fundamental
recentagle)
In a manner similar to the derivation of the equation of the ellipse, it can be shown using the
definition above that the equation of the ellipse with center (0,0) and principal axis the x-axis and
x2 y2
foci (c,0), (-c,0) is 2  2  1 where c 2  a 2  b 2 (illustrate with picture) Recall that for an
a
b
ellipse c 2  a 2  b 2
Similarly if the y-axis is the principal axis and the foci are (0,c) and (0,-c) the equation of the
y2 x2
hyperbola is 2  2  1 (illustrate with picture).
a
b
Note that regardless of whether the principal axis is the x or y axis, the length of the conjugate
axis is 2a and the length of the transverse axis is 2b. Note the use of the terms conjugate and
transverse axis for the hyperbola as opposed to the terms major axis and minor axis for the
ellipse. The major axis is always longer than the minor axis, as for an ellipse a > b. For the
hyperbola, note that a could be larger or smaller than b.
As for an ellipse, the eccentricity e = c/a.
Note that
c2 a2  b2
b

 1  
2
2
a
a
a
So the eccentricity of a hyperbola is a measure of the elongation of the fundamental rectangle. If
the fundamental rectangle is almost square then b is approximately equal to a and e2 is approx 2.
If the fundamental rectangle deviates greatly from being square then e2 is close to 1 in value or is
very large in value. Recall that for an ellipse, eccentricity is a number between 0 and 1.
2
e2 
Examples: p. 524 # 4,6, (graph and state coordinates of vertices, foci, equations of asymptotes,
eccentricity) 18,20
HW for Thursday, September 14, 2000
Read Section 13.4. Do # 1-23 odd, 27, 29