> ## Documentation Index
> Fetch the complete documentation index at: https://docs.iotools.cloud/llms.txt
> Use this file to discover all available pages before exploring further.
# Hyperbola Calculator
> Analyze a hyperbola in standard form: center, vertices, co-vertices, foci, eccentricity, and asymptotes, with a plotted diagram of both branches and step-by-step math.
[Try Hyperbola Calculator in your browser →](https://iotools.cloud/tool/hyperbola-calculator/)
## OpenAPI
````yaml https://api.iotools.cloud/v1/openapi post /v1/tool/hyperbola-calculator
openapi: 3.1.0
info:
title: iotools.cloud API
version: 1.0.0
description: >-
Run any iotools.cloud tool over HTTP.
Authenticate with `Authorization: Bearer iot_live_…`.
**Only `POST /v1/tool/{slug}` costs credits.** Every GET here — the catalog,
a tool's schema, your balance — is free. A tool call is charged its own
weight or your plan's per-call minimum, whichever is larger;
`x-iotools-credit-cost` on each operation is quoted at the free-tier
minimum, and `GET /v1/tools/list` returns the exact figure for your key.
`GET /v1/me/credits` reports what you have left, and `GET /v1/me/usage`
reports what it went on.
Failures are RFC 9457 problem documents — branch on `code`.
servers:
- url: https://api.iotools.cloud
description: Production
security:
- bearerAuth: []
tags:
- name: Catalog
description: Find a tool and read its contract. Free.
- name: Converters
description: >-
Convert between formats, encodings, and units — Base64, CSV and JSON,
timestamps, and more. Fast, free, and processed right in your browser.
- name: Formatters
description: >-
Format, beautify, minify, and validate code and data — JSON, HTML, CSS,
SQL, and regex. Clean up messy input in one click, with nothing to upload.
- name: Generators
description: >-
Generate exactly what you need and on demand — passwords, UUIDs, QR codes,
hashes, random numbers, and more. Secure, instant, and free.
- name: Calculators
description: >-
Crunch the numbers fast — from everyday math to specialized conversions
and unit work. Free online calculators that run entirely in your browser.
- name: Editors
description: >-
Edit and transform text, code, and images with quick, focused editors that
run entirely in your browser — nothing to install, and no sign-up needed.
- name: Utilities
description: >-
Everyday developer and web utilities — DNS and IP lookups, redirect and
certificate checkers, and other quick diagnostics. Free and
privacy-friendly.
- name: Account
description: Your key's allowance, limits and usage history.
paths:
/v1/tool/hyperbola-calculator:
post:
tags:
- Calculators
summary: Hyperbola Calculator
description: >-
Analyze a hyperbola in standard form: center, vertices, co-vertices,
foci, eccentricity, and asymptotes, with a plotted diagram of both
branches and step-by-step math.
[Try Hyperbola Calculator in your browser
→](https://iotools.cloud/tool/hyperbola-calculator/)
operationId: run_hyperbola_calculator
requestBody:
required: true
content:
application/json:
schema:
type: object
properties:
orientation:
enum:
- horizontal
- vertical
description: Orientation
a:
type: number
description: a
b:
type: number
description: b
h:
type: number
description: Center x (h)
k:
type: number
description: Center y (k)
required:
- a
- b
- h
- k
examples:
horizontal_centered_at_the_origin_a_3_b_2:
summary: Horizontal, centered at the origin (a = 3, b = 2)
value:
orientation: horizontal
a: '3'
b: '2'
h: '0'
k: '0'
vertical_off_center_a_2_b_4_center_1_2:
summary: Vertical, off-center (a = 2, b = 4, center (1, -2))
value:
orientation: vertical
a: '2'
b: '4'
h: '1'
k: '-2'
responses:
'200':
description: Tool output
content:
application/json:
examples:
horizontal_centered_at_the_origin_a_3_b_2:
summary: Horizontal, centered at the origin (a = 3, b = 2)
value:
tool: hyperbola-calculator
tool_version: 1.0.2
outputs:
svgSource: >-
summary:
- property: Orientation
value: Horizontal (opens left/right)
- property: Center (h, k)
value: (0, 0)
- property: a (transverse semi-axis)
value: '3'
- property: b (conjugate semi-axis)
value: '2'
- property: c (focal distance)
value: '3.60555'
- property: Eccentricity (e)
value: '1.20185'
- property: Vertices
value: (3, 0) and (-3, 0)
- property: Co-vertices
value: (0, 2) and (0, -2)
- property: Foci
value: (3.60555, 0) and (-3.60555, 0)
- property: Asymptotes
value: y = 0.666667x; y = -0.666667x
- property: Transverse axis length
value: '6'
- property: Conjugate axis length
value: '4'
steps: |-
Standard form: x² / 9 − y² / 4 = 1
Center: (h, k) = (0, 0)
Opens left/right (a = 3, b = 2)
c = √(a² + b²) = √(9 + 4) = 3.60555
Eccentricity: e = c / a = 3.60555 / 3 = 1.20185
Vertices: (3, 0) and (-3, 0)
Co-vertices: (0, 2) and (0, -2)
Foci: (3.60555, 0) and (-3.60555, 0)
Asymptotes: y = 0.666667x and y = -0.666667x
credits_used: 5
credits_remaining: null
vertical_off_center_a_2_b_4_center_1_2:
summary: Vertical, off-center (a = 2, b = 4, center (1, -2))
value:
tool: hyperbola-calculator
tool_version: 1.0.2
outputs:
svgSource: >-
summary:
- property: Orientation
value: Vertical (opens up/down)
- property: Center (h, k)
value: (1, -2)
- property: a (transverse semi-axis)
value: '2'
- property: b (conjugate semi-axis)
value: '4'
- property: c (focal distance)
value: '4.47214'
- property: Eccentricity (e)
value: '2.23607'
- property: Vertices
value: (1, 0) and (1, -4)
- property: Co-vertices
value: (5, -2) and (-3, -2)
- property: Foci
value: (1, 2.47214) and (1, -6.47214)
- property: Asymptotes
value: y = 0.5x - 2.5; y = -0.5x - 1.5
- property: Transverse axis length
value: '4'
- property: Conjugate axis length
value: '8'
steps: |-
Standard form: (y + 2)² / 4 − (x − 1)² / 16 = 1
Center: (h, k) = (1, -2)
Opens up/down (a = 2, b = 4)
c = √(a² + b²) = √(4 + 16) = 4.47214
Eccentricity: e = c / a = 4.47214 / 2 = 2.23607
Vertices: (1, 0) and (1, -4)
Co-vertices: (5, -2) and (-3, -2)
Foci: (1, 2.47214) and (1, -6.47214)
Asymptotes: y = 0.5x - 2.5 and y = -0.5x - 1.5
credits_used: 5
credits_remaining: null
schema:
type: object
required:
- tool
- tool_version
- outputs
- credits_used
- credits_remaining
properties:
outputs:
type: object
properties:
svgSource:
type: string
description: Diagram (SVG source)
summary:
type: array
description: Analysis
items:
type: object
properties:
property:
type: string
description: Property
value:
type: string
description: Value
x-iotools-columns:
- property
- value
steps:
type: string
description: Step-by-Step Solution
tool:
type: string
description: The tool's slug, echoing the {slug} in the request path.
tool_version:
type: string
description: Output-contract version for this tool.
credits_used:
type: integer
description: >-
Credits this call consumed, after any settlement refund. 0
when metering is disabled.
credits_remaining:
type:
- integer
- 'null'
description: >-
Credits left in the current monthly allowance, or null
when metering is disabled.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
'400':
$ref: '#/components/responses/ValidationError'
'401':
$ref: '#/components/responses/Unauthorized'
'402':
$ref: '#/components/responses/InsufficientCredits'
'403':
$ref: '#/components/responses/ToolNotAllowed'
'404':
$ref: '#/components/responses/ToolNotFound'
'413':
$ref: '#/components/responses/PayloadTooLarge'
'422':
$ref: '#/components/responses/UnprocessableInput'
'429':
$ref: '#/components/responses/RateLimited'
'500':
$ref: '#/components/responses/ToolFailed'
'503':
$ref: '#/components/responses/ToolDisabled'
security:
- bearerAuth: []
components:
responses:
ValidationError:
description: Invalid request.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/validation_error
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Invalid request
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- validation_error
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
fields:
type:
- object
- string
additionalProperties:
type: string
description: >-
What failed: a map of field name → message. Absent when the
body itself is malformed; a single string when the failure
isn't tied to one field.
example:
type: https://docs.iotools.cloud/errors/validation_error
title: Invalid request
status: 400
code: validation_error
detail: One or more inputs are invalid — see `fields`.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
fields:
inputString: Required
Unauthorized:
description: Missing or invalid API key.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/invalid_api_key
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Invalid API key
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- invalid_api_key
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
example:
type: https://docs.iotools.cloud/errors/invalid_api_key
title: Invalid API key
status: 401
code: invalid_api_key
detail: 'Provide ''Authorization: Bearer ''.'
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
InsufficientCredits:
description: Monthly credit allowance exhausted.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/insufficient_credits
- >-
https://docs.iotools.cloud/errors/vendor_api_tier_limit_exceeded
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Insufficient credits
- Free-tier API limit reached for this tool
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- insufficient_credits
- vendor_api_tier_limit_exceeded
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
credits_used:
type: integer
description: Always 0 — a refused call charges nothing.
credits_remaining:
type: integer
description: Credits left in the allowance — fewer than this call costs.
example:
type: https://docs.iotools.cloud/errors/insufficient_credits
title: Insufficient credits
status: 402
code: insufficient_credits
detail: This call costs 1 credit and 0 remain in this month's allowance.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
credits_used: 0
credits_remaining: 0
ToolNotAllowed:
description: Tool exists but has no API surface.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/tool_not_allowed
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Tool not available over the API
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- tool_not_allowed
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
example:
type: https://docs.iotools.cloud/errors/tool_not_allowed
title: Tool not available over the API
status: 403
code: tool_not_allowed
detail: >-
"Background Remover" is available on iotools.cloud but has no API
endpoint.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
ToolNotFound:
description: No such tool.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/tool_not_found
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Tool not found
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- tool_not_found
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
example:
type: https://docs.iotools.cloud/errors/tool_not_found
title: Tool not found
status: 404
code: tool_not_found
detail: No tool with that slug. See GET /v1/tools/list.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
PayloadTooLarge:
description: Body too large.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/payload_too_large
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Payload too large
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- payload_too_large
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
example:
type: https://docs.iotools.cloud/errors/payload_too_large
title: Payload too large
status: 413
code: payload_too_large
detail: Request body exceeds this tool's size limit.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
UnprocessableInput:
description: Well-formed request, but there is no result to return.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/unprocessable_input
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Input cannot be processed
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- unprocessable_input
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
credits_used:
type: integer
description: >-
Always 0 on `unprocessable_input` — a failed run is refunded,
floor included.
credits_remaining:
type: integer
description: >-
Credits left after the refund. Absent when metering is
disabled.
example:
type: https://docs.iotools.cloud/errors/unprocessable_input
title: Input cannot be processed
status: 422
code: unprocessable_input
detail: >-
This video has no captions or subtitles, so there is no transcript
to extract.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
credits_used: 0
RateLimited:
description: Per-minute rate limit exceeded.
headers:
Retry-After:
description: Seconds to wait before retrying (RFC 9110 delta-seconds).
schema:
type: integer
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/rate_limited
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Rate limit exceeded
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- rate_limited
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
retry_after:
type: integer
description: >-
Seconds until the window resets — the same value as the
`Retry-After` header.
example:
type: https://docs.iotools.cloud/errors/rate_limited
title: Rate limit exceeded
status: 429
code: rate_limited
detail: Too many requests. Retry in 30s.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
retry_after: 30
ToolFailed:
description: Tool failed to run.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/processing_error
- https://docs.iotools.cloud/errors/internal_error
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Tool failed to run
- Internal error
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- processing_error
- internal_error
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
credits_used:
type: integer
description: >-
Always 0 on `processing_error` — a failed run is refunded,
floor included.
credits_remaining:
type: integer
description: >-
Credits left after the refund. Absent when metering is
disabled.
example:
type: https://docs.iotools.cloud/errors/processing_error
title: Tool failed to run
status: 500
code: processing_error
detail: The tool failed to run. Please try again.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
credits_used: 0
ToolDisabled:
description: Tool temporarily disabled.
content:
application/problem+json:
schema:
type: object
description: >-
RFC 9457 problem document, served as application/problem+json.
Branch on `code`; `title` is human prose and may be reworded
without notice. Some failures add extension members — `fields` on
validation errors, `retry_after` on 429s,
`credits_used`/`credits_remaining` on billing-adjacent failures —
documented on the responses that carry them.
required:
- type
- title
- status
- code
properties:
type:
type: string
format: uri
description: Stable documentation URI for this failure.
examples:
- https://docs.iotools.cloud/errors/tool_disabled
- https://docs.iotools.cloud/errors/api_unconfigured
title:
type: string
description: Short human-readable summary of the failure.
examples:
- Tool temporarily disabled
- API not configured
status:
type: integer
description: HTTP status code, matching the response's own status.
code:
type: string
description: >-
Stable machine-readable error code — branch on this, not
`title`.
enum:
- tool_disabled
- api_unconfigured
detail:
type: string
description: Human explanation of this occurrence.
request_id:
type: string
description: Correlation id, also sent as x-request-id.
example:
type: https://docs.iotools.cloud/errors/tool_disabled
title: Tool temporarily disabled
status: 503
code: tool_disabled
detail: This tool is temporarily unavailable. Try again shortly.
request_id: e4042b29-8f1e-4c7a-9b52-6f0d1a3c7e11
securitySchemes:
bearerAuth:
type: http
scheme: bearer
bearerFormat: iot_live_…
````