Eccentricity

Eccentricity of a hyperbola is always greater than 1. e = √2 for a rectangular hyperbola (a = b). e → 1⁺ gives a very elongated hyperbola; e → ∞ gives one approaching a pair of lines. Directrices at x = ±a/e (between vertices and centre).

By the focus-directrix property, for a point $P(x_1, y_1)$ on the right branch of $x^2/a^2 - y^2/b^2 = 1$ with focus $S'(c, 0)$ and directrix $x = a/e$:

$$\frac{PS'}{PM} = e, \quad PS' = ex_1 - a, \quad PM = x_1 - \frac{a}{e}$$ $$\frac{ex_1-a}{x_1-a/e} = \frac{e(x_1-a/e)}{x_1-a/e} = e \checkmark$$

The eccentricity $e = c/a = \sqrt{1+b^2/a^2}$.

Since $b > 0$, we have $e > 1$ always.

Directrix location: The directrix $x = a/e$ lies between the centre and the vertex ($a/e < a$ since $e > 1$) — the opposite of the ellipse, where the directrix lies outside the curve.

Eccentricity of the conjugate hyperbola: For $x^2/a^2 - y^2/b^2 = -1$ (equivalently $y^2/b^2 - x^2/a^2 = 1$), eccentricity $e_2 = \sqrt{1+a^2/b^2} = c/b$.

The relation $\frac{1}{e_1^2} + \frac{1}{e_2^2} = 1$ holds between a hyperbola and its conjugate.