Figure 1
[Source: Wikipedia - Hyperbola]

The hyperbola is the most complicated of the three and, consequently, the most interesting. Figure 1 contains information about the hyperbola.
The vertices (\(\pm a\) on this plot) and the foci (F₁ and F₂ on this plot) lie on the transverse axis with the center at \((h,k)\). The standard equations are

As we did with the ellipse, we define an intermediate value \(c^2=a^2+b^2\) which will help us locate the foci.
For a hyperbola, we need to know the equations of the lines in blue on the plot. These are asymptotes.
For a horizontal transverse axis, the asymptotes are \(\displaystyle{ y=k \pm \frac{b}{a}(x-h) }\).
For a vertical transverse axis, the asymptotes are \(\displaystyle{ y=k \pm \frac{a}{b}(x-h) }\).
Similar to the ellipse, we define the eccentricity as \(e=c/a\). The results are summarized next.

Figure 2
[Source: Wikipedia - Hyperbola]

1. Notice that the value of c is different here than for an ellipse.
2. Since \(c > a\), \(e > 1\).

The hyperbola is quite a complicated graph with lots of features. Figure 2 gives you an idea of some of the other features that are involved with hyperbolas. Here is a video that goes into more detail.

Okay, time for some practice problems on hyperbolas. Although it may seem like we have given you a lot of information on this page, we have just skimmed the surface in the discussion of these three figures, especially the hyperbola. There are a lot more interesting features that we hope you get to explore in your class.