Hyperbola Calculator

For a hyperbola with semi-transverse axis $$a$$ and semi-conjugate axis $$b$$, the key relationships are:

Linear eccentricity: $$c = \sqrt{a^2 + b^2}$$

This differs from an ellipse where $$c = \sqrt{a^2 - b^2}$$. For a hyperbola, $$c > a$$ always, so the foci lie outside the vertices.

Eccentricity: $$e = \frac{c}{a} = \frac{\sqrt{a^2 + b^2}}{a}$$

Since $$c > a$$, we always have $$e > 1$$ for a hyperbola. Higher eccentricity means the branches are more open and the asymptotes are steeper.

Asymptotes: The two asymptote lines pass through the center with slopes $$\pm\frac{b}{a}$$, giving equations $$y - k = \pm\frac{b}{a}(x - h)$$. The hyperbola approaches these lines as $$x \to \pm\infty$$.

Semi-latus rectum: $$\ell = \frac{b^2}{a}$$, the distance from a focus to the curve measured perpendicular to the transverse axis.

Vertices are at $$(h \pm a, k)$$ and foci at $$(h \pm c, k)$$ for a horizontal hyperbola.

The eccentricity controls the shape: values near 1 produce a narrow hyperbola with nearly parallel branches, while large values produce wide-open branches. A rectangular hyperbola ($$a = b$$) has $$e = \sqrt{2} \approx 1.414$$.

The asymptote slopes define the "opening angle" of the hyperbola. The acute angle between the asymptotes is $$2\arctan(b/a)$$. When $$b > a$$, the hyperbola opens more widely than 90°.

The distance between foci ($$2c$$) determines the defining property: for any point on the hyperbola, the absolute difference of distances to the two foci equals $$2a$$.