Hyperbola vertices and directrix calculator
Enter the semi-axes 3 and 4 and the hyperbola gets vertices at (3, 0) and (−3, 0) and directrices at x = ±1.8, shown with c of 5 and eccentricity 5/3. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 3
- B
- 4
- C
- 5
- Eccentricity
- 1.6666666667
- Directrix x
- 1.8
- Directrices
- x = 1.8, x = -1.8
The vertices of a hyperbola are the two points where the curve meets its transverse axis, and the directrices are the pair of fixed lines used with the focus-directrix definition of the conic. For the standard horizontal form x squared over a squared minus y squared over b squared equals one, the vertices sit at plus or minus a on the x-axis, while the directrices are the vertical lines x equals plus or minus a over e, with eccentricity e equal to c over a and c equal to the square root of a squared plus b squared. This hyperbola vertices and directrix calculator accepts positive semi-axes a and b, returns both vertices, the directrix positions and equation strings, and the supporting c and e values, and rejects a zero semi-axis because the geometry degenerates and division would be undefined.
What hyperbola vertices and directrices are, and how a and b fix them
A hyperbola is the set of points whose absolute difference of distances to two fixed foci is constant, and equivalently the set of points whose distance to a focus is e times the distance to the corresponding directrix, with eccentricity e strictly greater than one. In the standard horizontal presentation centered at the origin, the locus is written x squared over a squared minus y squared over b squared equals one, where a is the semi-transverse axis and b is the semi-conjugate axis.
The vertices are the nearest points of the curve to the center along the transverse axis: they lie at coordinates a zero and minus a zero. The linear eccentricity c equals the square root of a squared plus b squared, so the foci sit farther out at plus or minus c on the x-axis. Eccentricity e equals c over a and is always greater than one for a genuine hyperbola.
The directrices are the vertical lines x equals plus or minus a over e, which is the same as plus or minus a squared over c. Those lines never meet the curve; they serve as the reference for the focus-directrix definition and appear in textbooks whenever eccentricity, latus rectum, or polar form of a conic is introduced.
Learners often confuse ellipse formulas where e is less than one and the directrix formula looks similar, swap a and b, or try to plot vertices when a typed zero collapses the transverse axis.
Where a hyperbola vertices and directrix calculator earns its keep
Classrooms and tutoring platforms use a focused hyperbola vertices and directrix calculator so learners can check that the vertices are at plus or minus a and that the directrices sit at plus or minus a over e without mixing ellipse formulas or forgetting the second directrix. Exam generators and LMS plugins need a deterministic oracle that marks a zero or negative semi-axis as invalid instead of silently taking an absolute value that would hide a student error.
CAD and technical-illustration scripts often need the vertex points and vertical directrix guides when sketching the curve next to foci and asymptotes from sibling tools. Physics and orbital-mechanics demos that already know the semi-axes can pull vertices and directrices without embedding ad-hoc square roots and divisions in every notebook cell. Failed validation does not consume balance.
Keep both semi-axes finite and strictly positive, treat the returned vertex coordinates and directrix strings as display-ready labels in your UI, and pin the rounded values in tests so homework checks, LMS plugins, and edge workers all share one reproducible answer without a computer-algebra system of your own.
What you can do with it
Homework and classroom checks
Students enter semi-axes a and b and verify vertices (±a, 0) and directrices x = ±a/e against textbook answers.
Exam and LMS auto-grading
Mark vertex and directrix items with a deterministic oracle that rejects zero or negative semi-axes and never invents a point.
FAQ
How do I find hyperbola vertices and directrices from the semi-axes?
Send positive numbers a (semi-transverse) and b (semi-conjugate).
What happens if a or b is zero?
A zero semi-axis makes the vertices or directrices undefined or degenerate; those failures are not billed.
Which form of the hyperbola is assumed?
The standard horizontal form x²/a² − y²/b² = 1 centered at the origin. Vertices are (±a, 0) and directrices are the vertical lines x = ±a/e.
Where are the foci of a hyperbola with semi-axes 3 and 4?
On the transverse axis at plus or minus c, where c is the square root of a squared plus b squared. With 3 and 4, c is 5, so the foci sit at (5, 0) and (−5, 0), farther out than the vertices.
Why do the directrices lie between the two vertices?
Each directrix sits at a divided by e, and a hyperbola always has an eccentricity greater than one, so a over e is smaller than a. With 3 and 4 the directrices fall at ±1.8, inside the vertices at ±3, and they never touch the curve.
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
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-vertices \
-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-vertices", {
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-vertices",
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-vertices", 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-vertices", 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_vertices",
"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
Published price — no tokens, no invented credits. A failed task is never charged.
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
Students, exam generators, and technical-drawing scripts can share one deterministic JSON result instead of re-deriving a over e by hand.
What hyperbola vertices and directrices are, and how a and b fix them
A dedicated hyperbola vertices and directrix calculator with validated positive inputs removes those slips while still exposing vertices, directrix coordinates, equation strings, and the intermediate c and e in one JSON object for the next step of a solution or pipeline.
Input contract, zero semi-axis errors, and response shape
Both semi-axes must be strictly positive. A zero semi-axis is rejected with a clear message because vertices would collapse to the origin or the eccentricity formula would divide by zero, and negative lengths are rejected because a semi-axis is a geometric length, not a signed coordinate. Those validation failures are not billed. Example: a equals three and b equals four yields c equals five, eccentricity five thirds, vertices at three zero and minus three zero, and directrices x equals one point eight and x equals minus one point eight. POST a JSON object with two required fields: a for the semi-transverse axis and b for the semi-conjugate axis. Aliases semi_transverse and semi_a for the first axis and semi_conjugate and semi_b for the second are accepted. Each value must be a finite number or a plain numeric string; missing, empty, non-numeric, NaN, and Infinity values fail with invalid_input naming the field. On success the body echoes a and b, reports c as the linear eccentricity, eccentricity as e equals c over a, vertices as the ordered pair of points at plus a and minus a on the x-axis, directrix_x as the positive distance a over e from the center, directrices as the two vertical equation strings, hyperbola_equation for the defining conic, orientation horizontal, center at the origin, and a short formula string. Keep requests tiny—two numbers in, a compact JSON object out—and pin the rounded vertices and directrix_x in automated tests so a future change to rounding would surface immediately.
Integration use cases
CAD and sketch pipelines
Feed measured semi-axes into the API and plot the two vertices plus the vertical directrix guides from the returned fields.
Technical questions
Is the hyperbola vertices and directrix calculator free?
The browser widget is free and runs locally. API calls use prepaid KIT balance at the published $0.002 base price; invalid input is not charged.
Is the result always the same for the same a and b?
Yes. The algorithm is pure analytic math with no randomness or timestamps, so identical inputs produce identical JSON.