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Hyperbola Asymptote Calculator

Enter the a and b values of your hyperbola to get its asymptotes, slope, and eccentricity.

How it is calculated

The tool assumes the standard hyperbola x²/a² − y²/b² = 1, centred at the origin and opening left and right. Here a is the distance from the centre to each vertex along the x-axis and b is the other semi-axis, which sets the rectangle around the centre. The asymptotes are the lines y = ±(b/a)x, so the slope is b/a. The eccentricity is e = √(1 + b²/a²), always above 1. Values are shown to four decimals. Both a and b must be positive; zero or a negative value shows no result. A hyperbola opening up and down (y²/a² − x²/b² = 1) or one not centred at the origin needs its equation rearranged first.

Worked example

For the hyperbola with a = 4 and b = 3: the slope is b / a = 3 / 4 = 0.75, so the asymptotes are y = 0.7500x and y = -0.7500x, shown with a slope of ±0.7500. For the eccentricity, b² / a² = 9 / 16 = 0.5625, so e = √1.5625 = 1.25, displayed as 1.2500. This is also c / a with c = √(4² + 3²) = 5, the distance from the centre to each focus, and 5 / 4 = 1.25 agrees.

Try this example →

Common mistakes

  • Taking a and b from the denominators of the equation instead of their square roots. In x²/16 − y²/9 = 1 the inputs are a = 4 and b = 3, not 16 and 9.
  • Swapping a and b. The slope is b / a, so entering them the wrong way round gives the reciprocal slope (1.3333 instead of 0.75).
  • Using the tool for a hyperbola that opens up and down, such as y²/9 − x²/16 = 1. There the slope is 3/4 (the root of the y-term's denominator over the x-term's), but entering 3 and 4 as a and b here would show 1.3333 instead.
  • Forgetting the centre. If the equation has terms like (x − 2), the asymptotes pass through that centre, not through the origin, and this tool shows only the centred form.
  • Assuming a larger eccentricity means a bigger hyperbola. It only describes the shape: a larger b / a ratio opens the branches wider.

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.

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