Setting the fous at the origin, i.e., , eq(2.22) simplifies to
which, in polar coordinates, becomes
where and [see Fig.2.13].
If X is on the same side of the directrix as the focus, so that (2.24) becomes
If X is on the other side of the directrix from the focus, and (2.24) becomes
Since, here, q is restricted to while , eq(2.26) cannot be satisfied for any q if . In other word, only hyperbola can have a branch on the other side of the directrix from the focus.
Consider now the branch
If , there is no restriction on q so that as befits an ellipse.
For , q is restricted to .