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Hyperbola Calculator: Foci, Vertices and Asymptotes

This free hyperbola calculator finds the foci, vertices, eccentricity, axis lengths and asymptote equations from the standard form, for hyperbolas that open sideways or up and down.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

A hyperbola has two branches and a pair of guide lines. Its focus and asymptote data are the usual homework and exam questions.

Choose the direction and enter the center, a and b.

How to Use This Tool

  1. Enter or paste your input in the box or boxes above.
  2. Choose the options, if there are any.
  3. Press the button to run the tool.
  4. Read the result and use the copy button where one is shown.

How the Calculation Works

The tool finds c = sqrt(a² + b²) and the eccentricity c / a. For a left-right hyperbola the vertices are (h ± a, k) and the foci (h ± c, k), and the asymptote slopes are ± b / a. For an up-down hyperbola the vertices are (h, k ± a), the foci (h, k ± c) and the slopes ± a / b.

Worked Example

For x²/9 - y²/16 = 1 (a = 3, b = 4) the foci are (-5, 0) and (5, 0), the eccentricity is 1.6667 and the asymptotes are y = ± 1.3333x.

Common Uses

  • Algebra and pre-calculus.
  • Navigation maths.
  • Optics and orbits.
  • Checking graphs.

Tips and Common Mistakes

  • Square the denominators to find a and b.
  • a is the one under the positive term.
  • The center defaults to (0, 0).
  • Check the first result against a case you already know, so you can trust the tool for the rest.
  • Nothing you type leaves your device. The work is done by your browser, so it also works with a slow connection once the page has loaded.

Next, you may need the parabola calculator, ellipse calculator, percentage calculator, average calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

a is the distance from the center to each vertex, b sets the width of the conjugate axis, and c is the distance to each focus, with c² = a² + b².
e = c / a. It is always greater than 1 for a hyperbola.
Lines the curve approaches but never meets. For a left-right hyperbola they are y - k = ± (b / a)(x - h).
The standard form with the center at (h, k). Move your equation to this form first.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.