Math
Hyperbola Graph (x²/a² − y²/b² = 1)
Enter a and b to graph the hyperbola x²/a² − y²/b² = 1. Automatically computes eccentricity, foci, vertices, and asymptotes.
Tips
- The hyperbola x²/a² − y²/b² = 1 consists of two separate branches. The right branch exists where x ≥ a and the left branch where x ≤ −a.
- The asymptotes y = ±(b/a)x are the lines the hyperbola approaches but never touches. They are shown as grey dashed lines on the graph.
- Eccentricity e = √(1 + b²/a²) is always greater than 1. Larger e values produce more "open" hyperbolas.
- The foci are at (±c, 0) where c = √(a² + b²). They appear as orange diamonds on the graph.
FAQ
The equation x²/a² − y²/b² = 1 requires |x| ≥ a, so there are no points near x = 0. The curve splits into a right branch (x ≥ a) and a left branch (x ≤ −a).
Asymptotes are the lines y = ±(b/a)x that the hyperbola approaches as x → ±∞. The curve never actually touches them, but gets arbitrarily close.
An ellipse is defined by the sum of distances to two foci being constant; a hyperbola uses the difference. An ellipse has eccentricity 0 < e < 1; a hyperbola has e > 1.
Side Note — Hyperbolas and navigation
A hyperbola is defined as the set of points where the difference of distances to two fixed foci is constant. This property powered the LORAN (Long Range Navigation) system used during the 20th century: ships located themselves by measuring the time difference between radio signals from two shore stations.
Hyperbolic mirrors appear in Cassegrain telescopes, where a convex hyperbolic secondary mirror reflects light through a hole in the primary mirror, placing the focal point behind the telescope body.