Prove an alternate formula for a conic section when the focus is in the positive half-plane

by RoRi

May 30, 2016

In Theorem 13.17 (page 501) of Apostol we established that a conic section $C$ with eccentricity $e$ , focus $F$ , and directrix $L$ at a distance $d$ from $F$ consists of all points $X$ satisfying

$\lVert X - F \rVert = e |(X-F) \cdot N - d|$

where $N$ is a unit normal to $L$ and $F$ is in the negative half-plane determined by $N$ . Prove that this formula must be replaced by

$\lVert X - F \rVert = e | (X-F) \cdot N + d|$

if $F$ is in the positive half-plane determined by $N$ .

Proof. Since $F$ is in the positive half-plane, we must have $(F-P) \cdot N = d > 0$ , so $d$ is positive. Then, replacing $P$ by $F-dN$ in the equation $\lVert X - F \rVert = e | (X-P) \cdot N |$ , we obtain