Eccentricity

Eccentricity of a Hyperbola

The eccentricity of a hyperbola is defined in the same spirit as it was for an ellipse — it measures how “stretched” the curve is. For a hyperbola, the eccentricity is the ratio of the distance from the centre to a focus, to the distance from the centre to a vertex.

Recall the standard hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (with centre at the origin). Its foci are at $(\pm c, 0)$ and its vertices are at $(\pm a, 0)$. The relationship between $a$, $b$, and $c$ is $c^2 = a^2 + b^2$.

This single number $e$ is called the eccentricity of the hyperbola.

Because $c \ge a$ (the foci are farther from the centre than the vertices), the eccentricity is never less than 1. For a hyperbola, $e > 1$ always. (If $e = 1$, the curve would degenerate into a parabola; if $e < 1$, it would be an ellipse.)

Expressing Foci in Terms of Eccentricity

Since $c = ae$, the foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ can be written directly using the eccentricity:

This is a compact way to locate the foci once $a$ and $e$ are known. Similarly, for the vertical hyperbola $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$, the foci are at $(0, \pm ae)$.

Key Idea: The Ratio $c/a$ Defines the Shape

The eccentricity is not just a label — it determines the shape of the hyperbola. For a fixed $a$, a larger $e$ means a larger $c$, which in turn means a larger $b$ (since $b^2 = c^2 - a^2$). The asymptotes become steeper, and the hyperbola opens more widely. …