In Theorem 13.17 (page 501) of Apostol we established that a conic section with eccentricity , focus , and directrix at a distance from consists of all points satisfying
where is a unit normal to and is in the negative half-plane determined by . Prove that this formula must be replaced by
if is in the positive half-plane determined by .
Proof. Since is in the positive half-plane, we must have , so is positive. Then, replacing by in the equation , we obtain