Hyperbola Calculator

For a hyperbola in standard position, x²/a² − y²/b² = 1, a is the semi-transverse axis (half the distance between the two vertices) and b is the semi-conjugate axis (half the distance between the co-vertices, which fix the curve's asymptotes but don't themselves lie on it). The distance from the center to each focus is c = √(a² + b²) — note the plus sign, the opposite of an ellipse's c² = a² − b², because a hyperbola's foci sit further out than its vertices rather than closer in.

Eccentricity, e = c/a, is always greater than 1 for a hyperbola (an ellipse's is always below 1, and a parabola's is exactly 1) — a larger e describes a 'wider,' flatter curve. The two asymptotes, the straight lines y = ±(b/a)x that the curve approaches but never touches, have slope exactly b/a. The semi-latus rectum, b²/a, is half the length of the chord that passes through a focus perpendicular to the transverse axis, a standard measure of how wide the curve is right at the focus.

Both axis lengths are treated as plain, unit-agnostic numbers — the same as this site's circle, ellipse, and triangle calculators — so the results come out in whatever unit the two axes were entered in.