🧲 Hyperbola Calculator
A hyperbola is the two-branched curve that appears in cooling towers, sonic-boom fronts and the paths of objects that escape a planet’s gravity. Where an ellipse is defined by a sum of distances, a hyperbola is defined by a difference — and it is the only conic that runs off to infinity along two straight guide lines, its asymptotes.
Choose the orientation, enter the centre and the two parameters a and b, and this calculator returns the vertices, foci, both asymptote equations and the eccentricity, with both branches drawn.
How to use this calculator
The hyperbola’s key relation is c² = a² + b² — a sum, unlike the ellipse’s difference.
- Choose the transverse axis: horizontal (opens left-right) or vertical (opens up-down).
- Enter the centre (h, k), a (centre to vertex) and b (conjugate semi-axis).
- Press Calculate.
- Read the vertices, foci and asymptotes, and watch the branches hug the dashed asymptotes on the graph.
Hyperbola Worked Example
A horizontal hyperbola centred at the origin with a = 3 and b = 2: the vertices are (±3, 0), and c = √(9 + 4) = √13 ≈ 3.61 puts the foci at (±3.61, 0). The asymptotes are the lines y = (2/3)x and y = −(2/3)x — the branches approach them ever more closely without ever touching. The eccentricity is e = √13/3 ≈ 1.20. These are the defaults above.
Frequently asked questions
What are the asymptotes of a hyperbola?
For a horizontal hyperbola centred at (h, k), the asymptotes are y − k = ±(b/a)(x − h). For a vertical one, y − k = ±(a/b)(x − h). The branches approach these lines as |x| or |y| grows, which is why sketching the asymptotes first is the standard way to draw a hyperbola by hand.
How is a hyperbola different from an ellipse?
Three things: the equation subtracts instead of adds (x²/a² − y²/b² = 1), the focal relation is c² = a² + b² instead of a difference, and the eccentricity is always greater than 1. Geometrically, an ellipse is a closed loop; a hyperbola is two open branches.
What does a and b mean in a hyperbola?
a is the distance from the centre to each vertex — half the transverse axis, the direction the branches open. b is half the conjugate axis; together a and b set the slope of the asymptotes (b/a for a horizontal hyperbola) and therefore how "open" the branches are.
Why is the eccentricity of a hyperbola always above 1?
Because c = √(a² + b²) is always larger than a, and e = c/a. e just above 1 gives narrow, pointed branches; large e gives wide, flat ones. It mirrors the ellipse, where e is always below 1.