This hyperbola calculator derives the full anatomy of a hyperbola — vertices, foci, asymptotes and eccentricity — from its center and the constants a and b.
| Equation | – |
| Vertices | – |
| Foci (c = √(a² + b²)) | – |
| Asymptotes | – |
| Eccentricity e = c/a | – |
c² = a² + b² — the Hyperbola Adds Where the Ellipse Subtracts
That plus sign is the whole personality difference from its conic sibling on our Ellipse Calculator (where c² = a² − b²). Part of our Math Calculators collection.
Worked Example (Verified)
- Horizontal, a = 3, b = 4, center origin (defaults): equation x²/9 − y²/16 = 1; c = 5; vertices (±3, 0); foci (±5, 0); asymptotes y = ±(4/3)x; e = 1.667
- The 3-4-5 triangle hiding in the defaults is why textbook hyperbolas love these values — c lands on an integer
- Vertical version of the same numbers: y²/9 − x²/16 = 1, vertices (0, ±3), foci (0, ±5), asymptotes y = ±(3/4)x — note the asymptote slope inverts when the opening flips
- Asymptotes are the skeleton: sketch the central rectangle (2a by 2b), draw its diagonals extended, and the branches hug them
Hyperbola vs Ellipse, Side by Side
- Equation sign: minus between terms (hyperbola) vs plus (ellipse).
- c relation: c² = a² + b² here; a² − b² for ellipses — foci live outside the vertices on a hyperbola.
- Eccentricity: always > 1 here; between 0 and 1 for ellipses.
- Defining distance property: |distances to foci| differ by constant 2a (hyperbola); they sum to 2a (ellipse).
Related Tools
- Ellipse Calculator — the closed conic
- Parabola Calculator — vertex, focus, directrix
- Math Calculators — the full collection
Frequently Asked Questions
How do you find the foci of a hyperbola?
c = √(a² + b²), placed c units from the center along the transverse axis. For a = 3, b = 4: c = 5, so a horizontal hyperbola at the origin has foci (±5, 0). Unlike ellipses, c exceeds a — the foci sit beyond the vertices.
What are a hyperbola’s asymptotes?
For a horizontal hyperbola: y − k = ±(b/a)(x − h); vertical swaps the ratio to ±(a/b). The defaults give y = ±(4/3)x. Branches approach these lines forever without touching — they are the graph’s guide rails.
What is the standard equation of a hyperbola?
Horizontal: (x − h)²/a² − (y − k)²/b² = 1. Vertical: (y − k)²/a² − (x − h)²/b² = 1. The positive term marks the transverse axis — a always sits under it, regardless of which letter is larger.
What does eccentricity mean for a hyperbola?
e = c/a, always greater than 1. Values just above 1 give narrow, pointed branches; large e opens them wide toward their asymptotes. The defaults’ e = 5/3 ≈ 1.667 is a middling, textbook-shaped hyperbola.
How is a hyperbola defined geometrically?
The set of points where the absolute difference of distances to two foci is constant (2a). Sound navigation (LORAN) exploited exactly this: a time difference between two transmitters places a ship on a hyperbola branch.
Is this hyperbola calculator free?
Yes — free, instant, no sign-up.