Consider again the central conics
Setting and , we have
which is called the standard form of the central conics. The foci are at . The directrices are vertical lines intersecting the x-axis at points .
For an ellipse with , we set so that (2.38) becomes
For a hyperbola with , we set so that (2.38) becomes
Now, as , (2.40) becomes
These give 2 lines with tangents that pass through the origin. They are called asymptotes of the hyperbola. See Fig.2.14.