Eccentricity

Eccentricity measures the deviation from a circle. e → 0 gives a circle (a = b); e → 1 gives a parabola in the limit (b → 0). Directrices are at x = ±a/e.

The eccentricity of a conic is defined as the ratio of the distance from any point on the conic to the focus, to the distance from the same point to the directrix:

$$e = \frac{PS}{PM}$$

For the ellipse $x^2/a^2 + y^2/b^2 = 1$ with focus $S(c, 0)$ and directrix $x = a/e$, let $P(x_1, y_1)$ be on the ellipse:

$$PS = |a - ex_1|, \quad PM = \left|\frac{a}{e} - x_1\right| = \frac{a - ex_1}{e} \quad (x_1 \leq a)$$ $$\frac{PS}{PM} = \frac{a - ex_1}{\frac{a-ex_1}{e}} = e \checkmark$$

For an ellipse, $0 < e < 1$:

• $e \to 0$: $c \to 0$, foci approach centre, ellipse approaches a circle
• $e \to 1$: $c \to a$, $b \to 0$, ellipse degenerates (in the limit it becomes a parabola)

Directrices: At $x = \pm a/e$ (farther from centre than vertices, since $a/e > a$).