How do you find the equation of a hyperbola?

Standard forms are (x − h)²/a² − (y − k)²/b² = 1 for horizontal hyperbolas and (y − k)²/a² − (x − h)²/b² = 1 for vertical hyperbolas. You need the center (h, k) and the semi-axes a and b.

How are foci of a hyperbola computed?

For both orientations c² = a² + b². For a horizontal hyperbola the foci are (h ± c, k); for a vertical one they are (h, k ± c).

What are the asymptotes of a hyperbola?

For the horizontal form (x − h)²/a² − (y − k)²/b² = 1 the asymptotes are y − k = ±(b/a)(x − h). For the vertical form (y − k)²/a² − (x − h)²/b² = 1 the asymptotes are y − k = ±(a/b)(x − h).

Analytic Geometry

Hyperbola Calculator

Build hyperbolas in standard or shifted form and get all key elements: center, vertices, foci, asymptotes and eccentricity. Inspired by eMathHelp and Symbolab, but simplified into a single panel.

Calculator

Hyperbola type

Decimal precision

a (semi-transverse)

b (semi-conjugate)

Results

Equation (standard form)

—

Center

—

Vertices

—

Foci

—

Eccentricity

—

Asymptotes

—

Hyperbola quick reference

Form	Asymptotes	Foci
\(\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1\)	\( y - k = \pm \dfrac{b}{a}(x - h) \)	\( (h \pm c, k) \), \(c = \sqrt{a^2 + b^2} \)
\(\dfrac{(y-k)^2}{a^2} - \dfrac{(x-h)^2}{b^2} = 1\)	\( y - k = \pm \dfrac{a}{b}(x - h) \)	\( (h, k \pm c) \), \(c = \sqrt{a^2 + b^2} \)

How this hyperbola calculator works

Classic analytic-geometry hyperbolas come in two main orientations: opening left–right (horizontal) and opening up–down (vertical). This tool covers both, plus the translated versions with center \((h, k)\).

1. Compute c and eccentricity

For all hyperbolas, \( c^2 = a^2 + b^2 \). Eccentricity is \( e = \dfrac{c}{a} \) and is always > 1.

2. Place vertices and foci

Horizontal: vertices at \((h \pm a, k)\), foci at \((h \pm c, k)\).
Vertical: vertices at \((h, k \pm a)\), foci at \((h, k \pm c)\).

3. Build asymptotes

Asymptotes are the most common thing people forget. The calculator writes them in point-slope form so you can graph quickly.

FAQ

What if a or b is zero?

Then the curve is not a proper hyperbola. The tool will warn you.

Can I go from general conic to hyperbola?

For that use our Conic Classifier first, then plug the simplified values here.

Audit: Complete

Formula (LaTeX) + variables + units

This section shows the formulas used by the calculator engine, plus variable definitions and units.

Formula (extracted LaTeX)

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Variables and units

No variables provided in audit spec.

Sources (authoritative):

NIST — Weights and measures — nist.gov · Accessed 2026-01-19
https://www.nist.gov/pml/weights-and-measures
FTC — Consumer advice — consumer.ftc.gov · Accessed 2026-01-19
https://consumer.ftc.gov/

Changelog

Version: 0.1.0-draft
Last code update: 2026-01-19

0.1.0-draft · 2026-01-19

Initial audit spec draft generated from HTML extraction (review required).
Verify formulas match the calculator engine and convert any text-only formulas to LaTeX.
Confirm sources are authoritative and relevant to the calculator methodology.

Verified by Ugo Candido on 2026-01-19
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