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THE HYPERBOLA A
hyperbola is a conic section with an
eccentricity greater than 1. The formulas
And
developed in the section concerning the ellipse were
derived so that they are true for any value of eccentricity. Thus, they are
true for the hyperbola as well as for an ellipse. Since e is greater than 1 for
a hyperbola, then
Therefore c > a > d.
Figure
217.The hyperbola. According to this analysis, if the center of symmetry of a
hyperbola is the origin, then the foci lies farther from the origin than the
directrices. An inspection of figure 217 shows that the curve never crosses
the Y axis. Thus the solution for the value of b, the
semiminor axis of the ellipse, yields no real value for b. In other words, b is an imaginary number. This can easily be seen
from the equation ^{} since c > a for a
hyperbola. However, we can square both sides of the the above
equation, and since the square of an imaginary number is a negative real number
we write
or _{
} and, since c = ae,
Now we can use this equation to obtain the equation of a
hyperbola from the following equation, which was developed in the section on
the ellipse: _{
} and since
we have _{
} This is a standard form for the equation of a hyperbola
with its center, O, at the origin. The solution of this equation for y gives
which shows that y is imaginary only when x^{2}
< a^{2}. The curve, therefore, lies entirely beyond the two
lines x = a and crosses the X axis at V_{1} (a,0) and V_{2}(
 a,0), the vertices of the hyperbola. The
two straight lines
can
be used to illustrate an interesting property of a hyperbola. The distance from
the line bx  ay = 0 to the point (x_{1},y_{1}) on the curve is given by _{
} Since
(x_{1},y_{1}) is on the curve, its coordinates
satisfy the equation
which
may be written
or
Now
substituting this value into equation (2.11)
gives us
As
the point (x_{1},y_{1}) is chosen farther and farther from
the center of the hyperbola, the absolute values for x, and y, will increase
and the distance, d, will approach
zero. A similar result can easily be derived for the line bx + ay = 0. The
lines of equation (2.10), which are usually written
are
called the asymptotes of the hyperbola. They are very important in tracing a
curve and studying its properties. The
Figure 218.Using asymptotes to sketch a hyperbola. asymptotes
of a hyperbola, figure 218, are the
diagonals of the rectangle whose center is the center of the curve and whose
sides are parallel and equal to the axes of the curve. The focal chord of
a hyperbola is equal to
. Another
definition of a hyperbola is the locus
of all points in a plane such that the difference of their distances from two
fixed points is constant. The fixed points are the foci, and the constant difference is 2a. The
nomenclature of the hyperbola is slightly different from that of an ellipse.
The transverse axis is of length 2a and is the distance between the
intersections (vertices) of the hyperbola with its focal axis. The conjugate axis is of length 2b and is perpendicular to the transverse
axis. Whenever
the foci are on the Y axis and the directrices are lines of the form y = k, where k is a constant, the equation of the hyperbola will read
This
equation represents a hyperbola with its transverse axis on the Y axis. Its
asymptotes are the lines by  ax = 0 and
by +ax=0 or
The properties of the hyperbola most often used in
analysis of the curve are the foci, directrices, length of the focal chord, and
the equations of the asymptotes. Figure 217 shows that the foci are given by the points F,
(c,0) and F_{Z} (  c,0) when the equation of the hyperbola is in the
form
If the equation were ^{
} the foci would be the points (0,c) and (0, c). The value
of c is either determined from the formula
or the formula
Figure 217 also shows that the directrices are the lines
or, in the case where the hyperbolas open
upward and downward,
. This was also given earlier in this discussion as
. 