Each hyperbola consists of two branches. The line segment which connects the two foci intersects the hyperbola at two points, called the vertices. The line segment which ends at these vertices is called the transverse axis and the midpoint of this line is called the center of the hyperbola. See Figure H1 for a sketch of a hyperbola with these pieces identified.

Note that, as in the case of the ellipse, a hyperbola can have a vertical or horizontal orientation.

We now turn our attention to the standard equation of a hyperbola. We say that the standard equation of a hyperbola centered at the origin is given by

Notice a very important difference in the notation of the equation of a hyperbola compared to that of the ellipse. We see that a always corresponds to the positive term in the equation of the ellipse. The relationship of a and b does not determine the orientation of the hyperbola. (Recall that the size of a and b was used in the section on the ellipse to determine the orientation of the ellipse.) In the case of the hyperbola, the variable in the “positive” term of the equation determines the orientation of the hyperbola. Hence, if the variable x is in the positive term of the equation, as it is in the equation

then the hyperbola is oriented as follows:

If the variable y is in the positive term of the equation, as it is in the equation

then we see the following type of hyperbola:

Note that the vertices are always a units from the center of the hyperbola, and the distance c of the foci from the center of the hyperbola can be determined using a, b, and the following equality:

We will use this relationship often, so keep it in mind.

The next question you might ask is this: “What happens to the equation if the center of the hyperbola is not (0,0)?” As in the case of the ellipse, if the center of the hyperbola is (h, k), then the equation of the hyperbola becomes

if the transverse axis is horizontal, or

if the transverse axis is vertical.

A few more terms should be mentioned here before we move to some examples. First, as in the case of an ellipse, we say that the eccentricity of a hyperbola, denoted by e, is given by

or we say that the eccentricity of a hyperbola is given by the ratio of the distance between the foci to the distance between the vertices. Now in the case of a hyperbola, the distance between the foci is greater than the distance between the vertices. Hence, in the case of a hyperbola,

Recall that for the ellipse,

Two final terms that we must mention are asymptotes and the conjugate axis. The two branches of a hyperbola are “bounded by” two straight lines, known as asymptotes. These asymptotes are easily drawn once one plots the vertices and the points (h, k + b) and (h, k – b) and draws the rectangle which goes through these four points. The line segment joining (h, k + b) and (h, k – b) is called the conjugate axis. The asymptotes then are simply the lines which go through the corners of the rectangle. (See FIGURE H4.)

But what are the actual equations of these asymptotes? Note that if the hyperbola is oriented horizontally, then the corners of this rectangle have the following coordinates:

and

Given two points, we can find the equation of the unique line going through the points using the point-slope form of the line. First, let us determine the slope of our line. We find this as “change in y over change in x” or “rise over run”. In this case, we see that this slope is equal to

or simply

Then, we also know that the line goes through the center

or

The other asymptote in this case has a negative slope which is given by

Using the same argument, we see that this asymptote has equation

What if the hyperbola is vertically oriented? Then one of the asymptote will go through the “corners”of the rectangle given by

Then the slope in this case will not be but will be . Hence, analogous to the work we just performed, we can show that the asymptotes of a vertically oriented hyperbola are determined by

and

Given all of this general information, we move to several examples.