Horizontal Hyperbola
(x−h)²/a² − (y−k)²/b² = 1
Vertical Hyperbola
(y−k)²/a² − (x−h)²/b² = 1
Distance to Foci
c = √(a² + b²)
Eccentricity
e = c/a (always > 1)
Center (h, k): The midpoint between the two branches
Transverse axis (a): Distance from center to each vertex
Conjugate axis (b): Determines the shape of the hyperbola
Foci: Two fixed points; the difference of distances from any point to the foci is constant (2a)
Asymptotes: Lines that the hyperbola approaches but never touches
A hyperbola is a type of conic section formed when a plane intersects both nappes (cones) of a double cone. It consists of two separate, mirror-image curves called branches. Unlike an ellipse, where the sum of distances to two foci is constant, a hyperbola is defined as the set of all points where the absolute difference of distances to two fixed points (foci) is constant and equals 2a.
Hyperbolas have many real-world applications including the paths of comets, the shape of cooling towers, navigation systems (LORAN), and the behavior of certain optical and radio wave systems. The distinctive asymptotic behavior makes hyperbolas useful in modeling phenomena that approach but never reach certain limits.
Hyperbola calculations follow standard mathematical formulas. Results depend on correct input values and axis configuration. This calculator is intended for educational purposes and should be verified for critical applications.