As in the book Apostol's Calculus saying:
Definition. Given a line L, a point F not on L, and a positive number e. Let $d(X,L)$ denote the distance from a point X to L.The set of ail X satisfying the relation $$ \lVert X-F \rVert = e\,d(X,L) $$ is called a conic section with eccentricity e. The conic is called an ellipse if $e < 1$, a parabola if $e = 1$, and a hyperbola if $e > 1$.
If N is a vector normal to L and if P is any point on L the distance d(X, L) from any point X to L is given by the formula: $$ d(X,L)=\frac{\lvert (X-P)\cdot N \rvert}{\lVert N \rVert} $$ When N has length 1, this simplifies to $d(X,L)=\lvert (X-P)\cdot N \rvert$. Then choose P to be that point on L nearest to F. Then $P-F=dN$ where $\lvert d \rvert=\lVert P-F \rVert$.
Replacing $P$ by $F+dN$ get the following theorem 13.17:
ThEOREM 13.17 Let C be a conic section with eccentricity e, focus F, and directrix L at a distance d from F. If N is a unit normal to L and if F is in the negative half-plane determined by N, then C consists of a11 points X satisfying the equation $$ \lVert X-F \rVert=e \lvert (X-F)\cdot N -d\rvert $$
"A set of points is said to be symmetric about the origin if $-X$ is in the set whenever X is in the set. We show next that the focus of an ellipse or hyperbola can always be placed so the conic section Will be symmetric about the origin. To do this we rewrite the basic equation (13.22) as follows:"
$$ \lVert X-F\rVert=e|(X-F)\cdot N - d| = |eX\cdot N - a| $$ where $a = ed+eF\cdot N$
Squaring both members, we obtain
(13.30)$$ \lVert X\rVert^2-2F\cdot N+\lVert F\rVert^2=e^2(X\cdot N)^2-2eaX\cdot N +a^2 $$
If we are to have symmetry about the origin, this equation must also be satisfied when X is replaced by -X, giving us
(13.31)$$ \lVert X\rVert^2+2F\cdot N+\lVert F\rVert^2=e^2(X\cdot N)^2+2eaX\cdot N +a^2 $$
Subtracting (13.31) from (13.30), we have symmetry if and only if $$ (F-eaN)\cdot X=0 $$
Then the book comes to a conclusion:
This equation can be satisfied for all X on the curve if and only if F and N are related by $$ F=eaN $$
Which I cannot understand. My question is under the condition that equation $(F-eaN)\cdot X=0$ always hold (where $X$ is on the conic section curve and $F-eaN$ can be considers as a constant vector) is it a must condition that $F-eaN=0$? What if $F-eaN$ be a vector which is not $0$ and is perpendicular to a plain where all vector $X$s on the conic curve. How it comes to the conclusion that $F-eaN=0$? Any hint will be appreciated. Thanks.
If I understand your notation correctly, $F - eaN$ must lie in the same plane as the conic section.
In fact, although Apostol, doesn’t say this, I think the entire construction that’s described here is assumed to be in the plane containing the point $F$ and the line $L$. If you drop this assumption, then even the original definition of a conic section (your first equation) is incorrect: the set of points $X$ in 3D space that satisfy $\| X - F \| = e\,d(X,L)$ isn’t a curve at all, it’s some sort of 3D quadric surface.