Focal Radii

For a point P(x₁, y₁) on the right branch (x₁ > 0) of x²/a² − y²/b² = 1: distance to nearer focus S′(c,0) is ex₁ − a, and to farther focus S(−c,0) is ex₁ + a. On the left branch (x₁ < 0), the nearer focus is S(−c,0) and the distances reverse.

For $x^2/a^2 - y^2/b^2 = 1$ with foci $S(-c,0)$ and $S'(c,0)$, directrices $x = -a/e$ and $x = a/e$.

Right branch ($x_1 > 0$):

Distance to $S'(c,0)$ — using directrix $x = a/e$:

$$r_2 = PS' = e \cdot PM' = e\!\left(x_1 - \frac{a}{e}\right) = ex_1 - a$$

Distance to $S(-c,0)$ — using directrix $x = -a/e$:

$$r_1 = PS = e \cdot PM = e\!\left(x_1 + \frac{a}{e}\right) = ex_1 + a$$

So $r_1 = ex_1 + a$ and $r_2 = ex_1 - a$, with $r_1 > r_2 > 0$ (since $ex_1 > a$ on the right branch because $x_1 \geq a$ and $e > 1$).

Left branch ($x_1 < 0$, so $|x_1| = -x_1$):

$$r_1 = PS = -ex_1 - a, \quad r_2 = PS' = -ex_1 + a$$

In all cases, the focal radii are always positive:

$$r_{\min} = a(e-1) \text{ at the vertex closer to a focus}$$ $$r_{\max} = a(e+1) \text{ at the vertex farther from a focus}$$