Hyperbola Asymptote Calculator

About this calculator

For a hyperbola in standard form x²/a² − y²/b² = 1, centered at the origin and opening left-right, the curve approaches but never touches two straight lines as x grows large — its asymptotes, given by y = ±(b/a)x. This calculator takes the a and b values and returns both asymptote equations, their slope b/a, and the hyperbola’s eccentricity e = √(1 + b²/a²), which measures how “open” or “narrow” the two branches are (e is always greater than 1 for a hyperbola, approaching 1 for a very narrow shape and growing larger as it flattens out).

Asymptotes are what make a hyperbola easy to sketch by hand — draw the rectangle with corners at (±a, ±b), extend its diagonals, and the branches hug those lines. Beyond analytic geometry coursework, this shape and its asymptotic behavior appear in orbital mechanics (a spacecraft on a hyperbolic trajectory that permanently escapes a planet’s gravity follows one branch, with the asymptote representing its eventual straight-line departure direction) and in the design of hyperbolic mirrors and lenses in optics.