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David Terr
Ph.D. Math, UC Berkeley

9.2. The Hyperbola

The hyperbola is the second of the three conic sections we discuss in this chapter. We already saw an example of a hyperbola in Section 1.6, namely Figure 1.6.2, the graph of the function y = 1/x. We will see in Section 9.4 that this graph is indeed a hyperbola, but for now, we discuss more general properties of hyperbolas.

The hyperbola is defined similarly to the ellipse. We start with two foci in the xy-plane, call them F1 and F2, as shown below. Let a be a positive number less than half the distance between the foci. Now consider the locus of all points P in the plane such that

• (9.2.1) |d(P, F1) - d(P, F2)| = 2a.

Figure 9.2.2: A hyperbola

(Image taken from MathWorld)

Note the similarity between Equation (9.1.1) for the ellipse and Equation (9.2.1) for the hyperbola. The only difference between these formulas is that (9.1.1) involves the sum of distances from a point on the figure to the foci and (9.2.1) involves the absolute value of the difference of these distances.

Note that the hyperbola shown above has two branches, one on the left and one on the right. In fact, all hyperbolas have two branches. One branch (the left branch in the above figure) corresponds to the difference d1 - d2 being negative, where d1 = d(P, F1) and d2 = d(P, F2), and the other corresponding to d1 - d2 being positive. The two branches of an ellipse are not connected, so every hyperbola is actually two separate curves.

The eccentricity of a hyperbola is defined in the same way as for an ellipse. The distance between the foci is defined to be 2c = 2ea, where once again, e is the eccentricity. Note that by this definition, the eccentricity of a hyprebola is greater than 1. (Eccentricity is defined for all four nondegenerate conic sections. A circle has eccentricity 0, an ellipse has eccentricity between 0 and 1, a parabola has eccentricity 1, and a hyperbola has eccentricity greater than 1.)

Another key feature of ellipses are its asymptotes. These are shown as dashed lines in the above figure. If we define b by the formula

• (9.2.3) b = √c2 - a2 = a√e2 - 1,

then the formula for the asymptotes is

• (9.2.4) y = ±(b/a)x.

As we will shortly see, graphing the asymptotes of a hyperbola greatly help us in graphing the hyperbola.

In order to be able to graph a hyperbola, we first need to know how to write down its equation. As it turns out, the equation for a hyperbola with its foci along the x-axis and centered at the origin is very similar to that of an ellipse with the same conditions, namely Equation (9.1.7). The formula is as follows:

• (9.2.5) x2 / a2 - y2 / b2= 1.

Note that Equations (9.1.7) and (9.2.5) differ only by the sign in the middle, (9.1.7) having a plus sign and (9.2.5) having a minus sign. However, this simple sign change gives rise to very different looking graphs!

So how do we graph a hyperbola given by Equation (9.2.5)? The first step is to draw with dashed lines the rectangle bounded by the vertical lines x = ±a and horizontal lines y = ±b. The next step is to draw, again with dashed lines, the two lines passing through each of the opposite pairs of vertices of this rectangle. These are the asymptotes of the hyperbola. The third step is to label the points P1 = (0, -a) and P2 = (0, a) on the hyperbola. These are the intersection points of the hyperbola and the x-axis. Finally, draw two curves, one passing through P1 and approaching the left sides of the asymptotes and the other passing through P2 and approaching the right sides of the asymptotes. Note that this procedure only requires that we plot two points on the hyperbola! Of course, we can draw a much more accurate graph if we have more points, which we should compute in any case.

Example 1: Graph the hyperbola given by the formula x2 - y2 = 4.

Solution: If we divide both sides of this formula by 4, it takes the form x2/4 - y2/4 = 1, which can also be written as x2 / 22 - y2 / 22= 1. Thus we see that a = b = 2. Our first step in graphing the hyperbola is to graph the square (shown as a dashed figure below) bounded by the lines x = ±2 and y = ±2.

Next, we connect opposite vertices of this rectangle with dashed lines as shown below. These lines are the asymptotes of the hyperbola.

Next, we label the points P1 = (0, -a) = (0, -2) and P2 = (0, a) = (0, 2) on the hyperbola.

Finally, we plot two smooth curves, one on the left passing through P1 and approaching the left sides of the asymptotes and the other passing through P2 and approaching the left sides of the asymptotes.

The graph you are likely to come up with will not be this accurate, but hopefully it will look something like this. To get a better graph, we need to plot more points. Note that we can solve for x in terms of y, obtaining

x = ±√4 + y2.

Letting y vary from -5 to 5 in intervals of 1 and solving for x, we obtain the following table:

x y
±5.39
-5.00
±4.47
-4.00
±3.61
-3.00
±2.83
-2.00
±2.24
-1.00
±2.00
0.00
±2.24
1.00
±2.83
2.00
±3.61
3.00
±4.47
4.00
±5.39
5.00

Plotting these points and following the above procedure yields the following graph, which is likely to be an improvement over the proceding one!

Figure 9.2.6: Graph of the hyperbola x2 - y2 = 4

It should be pointed out that graphing a hyperbola with equation y2 / b2 - x2 / a 2= 1 is just as easy to graph as the other type. This equation corresponds to a hyperbola with foci along the y-axis instead of the x-axis. Everything is the same as before except that the roles of x and y have been interchanged. Thus, the points P1 and P2 now lie on the y-axis instead of the x-axis and the two branches of the hyperbola are now on the top and on the bottom of the figure, instead of on the left and the right. The graph of the hyperbola y2 - x2 = 4 is shown below, along with the steps taken to produce it. Note that this hyperbola has the same asymptotes as the previous one.

Figure 9.2.6: Graph of the hyperbola y2 - x2 = 4