Our final conic will be the hyperbola. Recall that this is the two-branched curve we get when a plane that is parallel to the axis of a cone intercepts both nappes of that cone. It is also defined as follows:
A hyperbola is the set of all points in a plane such that the distances from each point to two 238 fixed points, the foci, have a constant difference.
Figure 22-89 shows the typical shape of the hyperbola. The point midway between the foci is called the center of the hyperbola. The line passing through the foci and the center is one axis of the hyperbola. The hyperbola crosses that axis at points called the vertices. The line segment connecting the vertices is called the transverse axis. A second axis of the hyperbola passes through the center and is perpendicular to the transverse axis. The segment of this axis shown in bold in Fig. 22-89 is called the conjugate axis. Half the lengths of the transverse and conjugate axes are the semitransverse and semiconjugate axes, respectively. They are also referred to as semiaxes.
We place the hyperbola on coordinate axes, with its center at the origin and its transverse axis on the x axis (Fig. 22-90). Let a be half the transverse axis, and let c be half the distance between foci. Now take any point P on the hyperbola ...