What is the difference between a and b in a hyperbola?

a is the semi-transverse axis — the distance from the center to each vertex along the axis of opening. b is the semi-conjugate axis, perpendicular to it. Both control the shape and the asymptote slopes (±b/a for horizontal, ±a/b for vertical).

How is c calculated for a hyperbola?

For a hyperbola, c² = a² + b² (note the plus sign, unlike an ellipse where c² = a² − b²). So c = √(a² + b²), and it gives the distance from center to each focus.

What does eccentricity greater than 1 mean?

Eccentricity e = c/a for a hyperbola is always > 1 because c > a. The larger e is, the more "open" or flat the hyperbola; as e approaches 1 from above, the branches narrow and approach a parabola.

What are the asymptotes of a hyperbola?

For a horizontal hyperbola x²/a² − y²/b² = 1, the asymptotes are y = ±(b/a)x. For a vertical hyperbola y²/a² − x²/b² = 1, they are y = ±(a/b)x. The branches approach but never touch these lines.

What is the latus rectum of a hyperbola?

The latus rectum is the chord through a focus perpendicular to the transverse axis. Its half-length is b²/a, so full length = 2b²/a. It measures the "width" of each branch at the focus.

Hyperbola Properties Calculator

Name: Hyperbola Properties Calculator
Availability: InStock
Author: Nham Vu

Enter a and b values for a standard-form hyperbola and instantly get vertices, foci, eccentricity, asymptotes, latus rectum, and more.

Hyperbola Parameters

Orientation

Horizontal
x²/a² − y²/b² = 1 Vertical
y²/a² − x²/b² = 1

a — semi-transverse axis

Distance center → vertex (positive)

b — semi-conjugate axis

Half-width of conjugate axis (positive)

Equation

x²/9 − y²/16 = 1

Core Properties

Vertices & Foci

Asymptotes & Directrices

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Summary

Enter a and b values for a standard-form hyperbola and instantly get vertices, foci, eccentricity, asymptotes, latus rectum, and more.

How it works

Select the orientation: horizontal (x²/a² − y²/b² = 1) or vertical (y²/a² − x²/b² = 1).
Enter the value of a (semi-transverse axis) — must be positive.
Enter the value of b (semi-conjugate axis) — must be positive.
The calculator derives c = √(a² + b²), eccentricity e = c/a, and all other properties.
Results update instantly; use the Copy button to grab any value.

Use cases

Check work on conic section homework or exams.
Verify hyperbola parameters when graphing on Desmos or a TI calculator.
Explore how changing a or b shifts vertices, foci, and asymptote angles.
Find directrix distances for optics and physics problems involving hyperbolic mirrors.

Frequently Asked Questions

Last updated: 2026-06-13 · Reviewed by Nham Vu