Mathematics · Geometry · Analytic Geometry
Hyperbola Calculator
Calculate the key properties of a hyperbola — including foci, vertices, asymptotes, eccentricity, and latus rectum — from its semi-major and semi-minor axes.
Calculator
Formula
For a horizontal hyperbola centred at the origin: a is the semi-transverse axis (distance from centre to each vertex), b is the semi-conjugate axis, c is the focal distance (distance from centre to each focus), e is the eccentricity (always > 1 for a hyperbola), and ℓ is the length of the latus rectum. Asymptotes are the lines y = ±(b/a)x.
Source: Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning. Section 10.5 — Conic Sections.
How it works
A hyperbola centred at the origin is defined by the standard equation x²/a² − y²/b² = 1 for a horizontal orientation, or y²/a² − x²/b² = 1 for a vertical orientation. The parameter a (semi-transverse axis) controls the distance from the centre to each vertex, while b (semi-conjugate axis) governs how steeply the branches flare outward. Unlike an ellipse, where c² = a² − b², the hyperbola obeys c² = a² + b², meaning the foci always lie farther from the centre than the vertices.
The eccentricity e = c/a is always greater than 1 for a hyperbola, distinguishing it from a circle (e = 0), an ellipse (0 < e < 1), and a parabola (e = 1). The asymptotes are straight lines y = ±(b/a)x that the hyperbola's branches approach but never touch; they pass through the centre and form a visual boundary for the curve. The latus rectum — the chord through each focus perpendicular to the transverse axis — has length 2b²/a and is useful when computing the shape of reflective surfaces and antenna cross-sections.
Practical applications span many fields. In orbital mechanics, a spacecraft performing a hyperbolic flyby traces one branch of a hyperbola around a planet. In navigation, LORAN and GPS systems historically used hyperbolic positioning: the difference in signal arrival times from two stations defines a hyperbola on which the receiver lies. Cooling towers at power plants are often shaped as hyperboloids (the 3-D solid of revolution of a hyperbola) for structural strength and airflow efficiency. Optics also uses hyperbolic mirrors in certain telescope designs (Cassegrain reflectors).
Worked example
Suppose a hyperbola has semi-transverse axis a = 5 and semi-conjugate axis b = 3, oriented horizontally. The standard equation is x²/25 − y²/9 = 1.
Step 1 — Focal distance: c = √(a² + b²) = √(25 + 9) = √34 ≈ 5.8310 units. The foci are located at (±5.8310, 0).
Step 2 — Eccentricity: e = c/a = 5.8310 / 5 ≈ 1.1662. Since e > 1, this confirms a hyperbola.
Step 3 — Asymptotes: The slope magnitude is b/a = 3/5 = 0.6000. The asymptote equations are y = 0.6x and y = −0.6x.
Step 4 — Latus rectum: ℓ = 2b²/a = 2(9)/5 = 3.6000 units. Each focus has a chord of length 3.6 units running perpendicular to the transverse axis.
Step 5 — Axis lengths: Transverse axis = 2a = 10 units; Conjugate axis = 2b = 6 units.
Limitations & notes
This calculator assumes the hyperbola is centred at the origin with axes aligned to the coordinate axes (no rotation or translation). For hyperbolas of the form (x−h)²/a² − (y−k)²/b² = 1, the same formulas apply but the foci and vertices are offset by (h, k). The calculator does not handle rotated conics (xy cross-terms), which require diagonalisation of the conic matrix. All inputs must be strictly positive real numbers; entering a = 0 or b = 0 produces a degenerate conic. This calculator handles only real hyperbolas — if the equation yields imaginary axes, the conic is an imaginary hyperbola and the results lose physical meaning.
Frequently asked questions
What is the difference between the semi-transverse axis and the semi-conjugate axis?
The semi-transverse axis (a) is the distance from the centre of the hyperbola to each vertex along the axis of symmetry that passes through the foci. The semi-conjugate axis (b) is a perpendicular half-length that, together with a, defines the shape of the asymptotes and the steepness of the hyperbola's branches. Neither axis is inherently the 'larger' one — unlike an ellipse, a and b can take any positive values relative to each other.
Why is the eccentricity of a hyperbola always greater than 1?
Eccentricity e = c/a, and for a hyperbola c = √(a² + b²), so c > a always (since b > 0). Therefore e = c/a > 1 for every non-degenerate hyperbola. Eccentricity encodes how 'open' the branches are: values just above 1 give a nearly parabolic shape, while very large e values produce nearly straight branches.
How do I find the equations of the asymptotes from a and b?
For a horizontal hyperbola centred at the origin, the asymptote equations are y = (b/a)x and y = −(b/a)x. For a vertical hyperbola, they are y = (a/b)x and y = −(a/b)x. These lines form the diagonals of the rectangle with half-sides a and b, and the hyperbola's branches approach but never cross them.
What is the latus rectum of a hyperbola and why is it useful?
The latus rectum is the chord drawn through a focus perpendicular to the transverse axis. Its full length is 2b²/a. It is useful in optics and antenna design because it characterises the curvature of the hyperbola near each focus — the tighter the curvature, the shorter the latus rectum relative to the focal length.
Can a hyperbola have equal values of a and b?
Yes — when a = b the hyperbola is called a rectangular (or equilateral) hyperbola. Its asymptotes are perpendicular to each other (slopes ±1), and the equation simplifies to x² − y² = a². Rectangular hyperbolas appear frequently in economics (e.g., the demand curve PQ = constant) and in complex analysis.
Last updated: 2025-01-15 · Formula verified against primary sources.