|Let F1and F2 be fixed points in the plane. A hyperbola is the collection of points in the plane such that the difference of the distances from the point to F1and F2 is a fixed constant. The points F1and F2 are called the foci of the hyperbola.|
|The picture illustrates the definition. The point P is a typical point
on the hyperbola. The difference of the distances from P to each of the foci is a constant. If we denote this constant by 2a, a > 0, then
Other than the foci
there are other special points associated with a hyperbola
which we have pointed out in the diagram.
The line through the foci cuts the hyperbola at two points, labeled
A and B in the diagram.
The points A and B are called the vertices.
The line segment joining the vertices
is called the transverse axis.
The point midway between the foci and lying on the
transverse axis is called the center of the
hyperbola. This is the point labeled O in the diagram.
If we position a hyperbola in the plane with its center at the origin and its foci along the x axis we can obtain a nice equation for a hyperbola. The derivation of the equation is almost identical to the derivation of the equation for the ellipse. We will not go through all the details here but set up the machinery for you to obtain the equation. As with the ellipse, we let the positive constant be 2a and let c be a positive number so that the foci are located at (-c,0) and (c,0). We have
To understand the importance of the asymptotes let's rewrite the equation of the hyperbola as x2/a2 = y2/b2 + 1. Now for very large values of x the addition of 1 on the right side of the equation becomes insignificant. That is, for very large values of x
The numbers a and b do have geometric meaning. Referring to the diagram,
we know that at the points A and B, y=0. Then, substituting
y=0 in the equation of the hyperbola gives
x2/a2 = 1 which simplifies to
x2=a2. This equation has solutions
x=a or -a. Since a > 0, then A has coordinates (-a,0) and B has coordinates
(a,0). Thus the vertices of the hyperbola are at (-a,0) and (a,0).
Now draw in points, C and D, with coordinates (0,b) and (0,-b) respectively.
The line segment CD passes through the center, O, and
is perpendicular to the transverse
axis. It is called the
Note that the diagonals of the rectangle containing A, B, C and D are line
segments of the asymptotes. This gives a convenient way to draw in the
asymptotes - namely draw vertical lines through the vertices, draw horizontal
lines through the points (0,b) and (0,-b). The points of intersection of these
four lines form the vertices of a rectangle. Draw in the lines that contain the
diagonals of the rectangle and you have drawn the asymptotes.
Note that the length of CD is 2b and the length of AB is 2a (compare with the
minor and major axis of the ellipse).
Note that since
b2 = c2-a2
then b < c and similarly a < c.
In a similar manner, we could have placed the foci on the y axis and the center at the origin In this case, the equation of the hyperbola comes out to be
The equations we have just established are known as standard equations of a hyperbola in standard position. Standard position always implies the center is at the origin and the foci are on one of the axes.
|Hyperbolas in standard position|
|Example||Find the foci of the hyperbola
The foci are located on the x axis since the x term is positive. We have a2 = 16 and b2 = 9. Then c2 = 25 and so c = 5. Therefore, the foci are located at (-5,0) and (5,0).
|Example||Find the equation of the hyperbola in standard position
with a focus at (0,13) and
with transverse axis of length 24.
The other focus is located at (0,-13) and since the foci are on the y axis we are looking to find an equation of the form y2/a2-x2/b2 = 1. The value of a is one-half the length of the transverse axis and so a = 12. Also, b2 = c2 - a2 = 169 - 144 = 25. Hence b = 5. So the equation of the hyperbola is
|Equation of a hyperbola|
If we translate a hyperbola in standard position so that its center is moved to (h, k) then the equation of the hyperbola is given as follows:
The equation of a hyperbola translated from standard position so that its
center is at (h,k) is given by
This form of the equation of a hyperbola is called the standard
equation. However, if we take a standard equation
(x-h)2/u2-(y-k)2/v2 = 1
and expand it we get an equation of the form
is the equation of a hyperbola. If so, find the coordinates of
the center and the foci.
Our first task is to group together the x terms and group together the y terms. We have
(16x2 -64x) -(9y2 -18y)-89=0
16(x2 -4x) -9(y2 -2y)-89=0
16((x-2)2 -4) -9((y-1)2 -1)-89=0
16(x-2)2 -9(y-1)2 -144=0
16(x-2)2 -9(y-1)2 =144
(x-2)2/9 -(y-1)2/16 =1
This we should recognize as a hyperbola with center (2,1). The transverse axis is parallel to the x axis. Now, c2 = 9 + 16 = 25, so that c = 5. The foci of the hyperbola in standard position (and foci along the x axis) are at (c,0) and (-c,0) so the foci of the translated hyperbola are at (c+2,1) and (-c+2,1). That is, at (7,1) and (-3,1).
|Finding the Center and Foci of a Hyperbola|