Hyperbola
The two-branched conic with a constant difference of focal distances, hugging its asymptotes forever.
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Formulas and key values
- Cartesian equation
- Asymptote
- Foci
- EccentricityIn the graph ≈ 1.329
a is the center-to-vertex distance, b sets the asymptote slope, and c is the center-to-focus distance.
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Start the video01 · Definition
What is the Hyperbola?
The hyperbola is the conic with eccentricity greater than 1: the set of points whose focal distances differ by a constant 2a. Unlike the ellipse it splits into two branches, and unlike any other conic it comes with a pair of built-in guide lines — the asymptotes y = ±(b/a)x that the branches approach but never touch.
Constant difference of distances is exactly what a time difference of arrival measures, which made hyperbolas the mathematics of LORAN radio navigation before GPS. The same curve appears as y = 1/x in disguise (a rotated rectangular hyperbola), as the shadow of a sundial tip through a day, and as the profile of power-plant cooling towers.
02 · Properties
Key properties
- Foci at (±c, 0) with c² = a² + b²; eccentricity e = c/a > 1.
- Difference of focal distances is constant: |r₁ − r₂| = 2a.
- Asymptotes y = ±(b/a)x; a = b gives the rectangular hyperbola (y = 1/x rotated 45°).
- Reflective property: rays aimed at one focus reflect toward the other.
- Conjugate hyperbola swaps the roles of the two axes.
03 · Questions
Hyperbola: common questions
- What is the equation of the Hyperbola?
- Cartesian equation: (x^2)/(a^2) - (y^2)/(b^2) = 1. a is the center-to-vertex distance, b sets the asymptote slope, and c is the center-to-focus distance.
04 · Labs
Interactive labs
Conic sections and eccentricity
How can one number turn a circle into an ellipse, a parabola and a hyperbola?
Focus and directrix · Eccentricity · Ellipse, parabola, hyperbola
Open labInterference of two waves: path difference decides
Two speakers play the same note. Why is it loud in some spots and almost silent in others?
Superposition · Path difference · Constructive and destructive
Open labThe hyperbola: a constant difference of distances, and its asymptotes
What stays constant on a hyperbola, and why do its branches hug the lines y = ±(b/a)x without touching them?
Definition by two foci · Standard equation · c² = a² + b²
Open lab05 · Keep exploring