ForHosting KIT · Math

Hyperbola calculator

Enter a hyperbola's two semi-axes and get all its elements: semi-axes 3 and 4 give foci at ±5, vertices at ±3, an eccentricity of 5/3 and asymptote slopes of ±4/3. Ready for answer keys and sketches. Free in your browser.

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Run — free How it works & API

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Worked example

If you enter

Semi-transverse axis: 3
Semi-conjugate axis: 4

You get

C: 5
A squared: 9
B squared: 16
C squared: 25

A rectangular-coordinate hyperbola is defined by two positive semi-axes: a along the transverse axis and b along the conjugate axis. From those two numbers you recover the linear eccentricity c as the square root of a squared plus b squared, place the vertices at plus and minus a on the x-axis, place the foci at plus and minus c, form the eccentricity e equals c over a (always greater than one for a non-degenerate hyperbola), and read the asymptote slopes as plus and minus b over a.

What the semi-axes mean and how the formulas work

Start from the standard horizontal equation x squared over a squared minus y squared over b squared equals one, with the center at the origin and the transverse axis along x. The parameter a is the semi-transverse axis: each vertex sits at (a, 0) and (−a, 0), so the distance from the center to a vertex is exactly a. The parameter b is the semi-conjugate axis; together a and b build the rectangle that touches the asymptotes and fixes the opening of the branches.

The linear eccentricity is c equals the positive square root of a squared plus b squared—always larger than a when b is positive, which forces the eccentricity e equals c over a to exceed one. The two foci lie at (c, 0) and (−c, 0). The asymptotes are the lines y equals (b over a) x and y equals minus (b over a) x, so the reported asymptote slopes are simply plus and minus b over a.

A classic check is a equals three and b equals four: then c equals five, e equals five thirds, vertices sit at plus and minus three on the x-axis, foci at plus and minus five, and slopes are plus and minus four thirds. Every step is elementary arithmetic and one square root on two positive scalars, so there is no iteration, no sampling, and no dependence on wall-clock time.

What you can do with it

Precalculus and analytic geometry lesson keys

Generate answer-key numbers for horizontal hyperbolas without a CAS, including equation strings and slope pairs for worksheets.

Engineering sketches needing foci and asymptotes

Quickly recover interfocal distance and asymptote slopes when a design brief only lists the two semi-axes of a hyperbolic branch.

FAQ

What inputs does the hyperbola calculator need?

Two positive numbers: a (semi-transverse axis) and b (semi-conjugate axis).

How are the foci computed?

First form c = sqrt(a² + b²). For the standard horizontal hyperbola the foci are at (c, 0) and (−c, 0).

What is the eccentricity of a hyperbola?

Eccentricity e = c/a with c = sqrt(a² + b²). For any positive a and b, e is strictly greater than 1.

What are the asymptote slopes?

For x²/a² − y²/b² = 1 the asymptotes are y = ±(b/a)x, so the slopes are +b/a and −b/a.

Is the hyperbola calculator free?

The browser tool is free and runs locally.

Related tools

For developers — API access

Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.

API endpoint

POSThttps://api.kit.forhosting.com/geom/hyperbola

Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.

Call it from your stack

curl -X POST https://api.kit.forhosting.com/geom/hyperbola \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"a":3,"b":4}'

const res = await fetch("https://api.kit.forhosting.com/geom/hyperbola", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "a": 3,
    "b": 4
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/geom/hyperbola",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "a": 3,
    "b": 4
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/geom/hyperbola", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"a":3,"b":4}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"a":3,"b":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/geom/hyperbola", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "a": 3,
  "b": 4
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "geom.hyperbola",
  "status": "queued",
  "_links": {
    "result": "/tasks/tsk_…/result"
  }
}

The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.

Pricing

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Limits

`max_abs`	1000000000000
`min_positive`	1e-15

Errors

HTTP	Code	Meaning
`401`	`unauthorized`	Missing or invalid API key.
`402`	`insufficient_balance`	Your balance doesn't cover the task price.
`404`	`unknown_type`	That task type doesn't exist.
`429`	`rate_limited`	Too many requests. Use the webhook instead of polling.

Technical notes

This capability packages that closed-form path into one deterministic JSON response so students checking homework, teachers writing answer keys, and engineers verifying orbital or optical asymptote geometry all get the same rounded figures in the free browser tool and on the prepaid API path—no computer-algebra system, no random seeds, and no network call inside the solver.

How to call the API and what comes back

Use it from analytic-geometry homework pipelines, precalculus review scripts, interview-prep sandboxes, optics and radio-range sketches that need asymptote slopes, or any workflow that needs reproducible hyperbola elements without installing a computer-algebra package on every machine. Send a JSON body with two required fields. Field a is the positive semi-transverse axis; aliases semi_transverse and semi_a are accepted. Field b is the positive semi-conjugate axis; aliases semi_conjugate and semi_b map to the same meaning. On success the response echoes a and b, adds c, a_squared, b_squared, c_squared, eccentricity, vertices as two points, foci as two points, asymptote_slopes as the pair plus and minus b over a, human-readable asymptote strings, the standard equation string, orientation horizontal, transverse_axis x, center at the origin, and a short formula block documenting the closed forms. Failed validation—missing a required field, a non-finite number, a non-positive semi-axis, a value outside published limits, or a recovered c that overflows—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.

Practical tips, limits, and related searches

When teaching, prefer paired examples: three and four for the Pythagorean c equals five check; five and twelve for another integer c; and a non-positive trial such as a equals zero or b negative to demonstrate the hard rejection path that protects callers from a degenerate figure. Prefer this tool when the textbook gives semi-axes a and b for the horizontal form and you need foci, vertices, e, and slopes in one shot. Round-trip checks are easy: after solving, confirm that c squared equals a squared plus b squared, that e equals c over a, and that the asymptote slopes equal plus and minus b over a within floating tolerance. Keep a and b inside the published positive range up to one trillion so intermediate squares and the sum under the radical stay finite in IEEE double precision on the Worker and in the browser. Units are whatever you pass—metres, astronomical units, or pure abstract units—as long as a and b share the same unit; the API does not convert units for you. Because eccentricity is always greater than one for positive a and b, students can recompute c and e by hand for small integers and match the API line by line. This landing targets people searching for a hyperbola calculator, foci and vertices tool, or asymptote slope helper; it is not an ellipse calculator—ellipses use c squared equals a squared minus b squared with eccentricity less than one. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when users type impossible semi-axes while learning the contract.

Integration use cases

Homework checks for standard hyperbola elements

Give students semi-axes a and b; compare their hand-computed c, eccentricity, vertices, foci, and asymptote slopes against a deterministic API or free browser result.

Read the full KIT documentation →

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