How to use the hyperbola calculator
A hyperbola has two standard forms. The horizontal form x²/a² − y²/b² = 1 opens left and right, while the vertical form y²/a² − x²/b² = 1 opens up and down. The orientation decides which axis holds the vertices and foci and also changes the asymptote slopes, so pick the form first and then enter a and b.
The focal distance is c = √(a² + b²) for both forms. An ellipse uses a² − b², but a hyperbola adds the two squares, which is the detail most often missed. The asymptotes are y = ±(b/a)x for the horizontal form and y = ±(a/b)x for the vertical form. When a and b are whole numbers the slope is reduced to a fraction and a decimal approximation is shown next to it.
The results list the standard form, the vertices, the foci, the focal distance c, the asymptotes, the eccentricity e = c/a, the transverse axis 2a, the conjugate axis 2b, the distance between foci 2c and the latus rectum 2b²/a. When c is irrational it appears both in exact radical form, such as √13, and as a decimal with 8 significant digits. Because a and b are lengths, zero or negative values are rejected with a message. Hyperbolas whose center is shifted away from the origin are not supported.
Frequently Asked Questions
Use c = √(a² + b²). A horizontal hyperbola has foci at (±c, 0) and a vertical one at (0, ±c). Unlike an ellipse, a² and b² are added.
For x²/a² − y²/b² = 1 the asymptotes are y = ±(b/a)x, while for y²/a² − x²/b² = 1 they are y = ±(a/b)x, because the letter in the denominator swaps.
For a hyperbola c is always larger than a, so e = c/a exceeds 1. Values near 1 give a narrow opening and larger values a wider one.