What is Focus–directrix definition?

Example 24.12 (The definition, checked on a parabola)

Take the parabola $y^2 = 4x$, i.e. $2p = 4$: focus $F = (1, 0)$ and directrix $D : x = -1$ (the reduced form Theorem 24.13 puts them at $\pm\frac p2$). At the point $M = (1, 2)$ of the curve:

equal, as $e = 1$ demands. At $M' = (4, 4)$: $MF' = \sqrt{9 + 16} = 5$ and $d(M', D) = 5$ again. The focus–directrix definition is not an abstraction: it is a pair of distances one can measure on any point, and the algebra of the reduced equations is nothing but this measurement done once and for all.

Example 24.15 (Reading an orbit from its polar equation)

The polar conic $r = \dfrac{1}{1 + \frac12\cos\theta}$ (Exercise 24.7 with $p = 1$, $e = \frac12$) is an ellipse with a focus at the origin — the geometry of a planetary orbit with the sun at $O$. Extract everything from $p$ and $e$: from the reduction of Theorem 24.13, $a = \dfrac{p}{1 - e^2} = \dfrac{1}{3/4} = \dfrac43$ and $c = ea = \dfrac23$. The two apsides check it with no theory at all:

and their sum $\frac23 + 2 = \frac83 = 2a$ recovers the major axis. The polar form is the natural one whenever a focus is physically distinguished; the reduced cartesian form, whenever the symmetry axes are. Converting between the two is exactly what the completed-square computation of the theorem does.

Eccentricity as a dial, finally: keep $p = 1$ and turn $e$ in $r = \frac{1}{1 + e\cos\theta}$. At $e = 0$: the circle $r = 1$. At $e = \frac12$: the ellipse just studied, $r$ oscillating between $\frac23$ and $2$. At $e = 1$: $r(\theta) \to \infty$ as $\theta \to \pi$ — the curve no longer closes: a parabola, its farthest point pushed to infinity. At $e = 2$: the denominator vanishes at $\cos\theta = -\frac12$, and only $\theta \in \intoo{-\frac{2\pi}3}{\frac{2\pi}3}$ survives: one branch of a hyperbola, escaping along two asymptotic directions. One formula, the entire conic family, and the transition points $e = 1$ visible as the moment the denominator first reaches zero.