Relation Between a, b, and c

Unlike the ellipse where c² = a² − b², for a hyperbola c² = a² + b² (the foci are farther from the centre than the vertices). b² is not less than a²; a and b have no size constraint relative to each other.

From the derivation of the standard form, $b^2 = c^2 - a^2$, equivalently:

$$c^2 = a^2 + b^2$$

Geometric meaning: Unlike the ellipse where the focal distance $c$ is shorter than the semi-major axis $a$, for a hyperbola $c > a$ always. The foci lie outside the curve (beyond the vertices).

Contrast with other conics:

In terms of eccentricity: $c = ae$, so:

$$b^2 = c^2 - a^2 = a^2e^2 - a^2 = a^2(e^2-1)$$

Since $e > 1$ for a hyperbola, $b^2 > 0$ — consistent. As $e \to 1^+$, $b \to 0$ (degenerate). As $e \to \infty$, the asymptotes approach the transverse axis.