Compute all essential properties of a hyperbola: foci, vertices, asymptotes, eccentricity, latus rectum, and center. Visualize the curve, asymptotes, and key points on an interactive canvas. Supports horizontal and vertical orientations.
A hyperbola is the set of all points such that the absolute difference of distances to two fixed points (foci) is constant. It consists of two disconnected branches and is one of the four conic sections, alongside circles, ellipses, and parabolas. The standard equation for a horizontal hyperbola centered at (h,k) is ((x-h)² / a²) - ((y-k)² / b²) = 1; for vertical, ((y-k)² / a²) - ((x-h)² / b²) = 1. Here, a is the distance from center to each vertex (semi-transverse axis), b the semi-conjugate axis, and c = √(a² + b²) the distance to each focus.
c² = a² + b² e = c/a (e > 1) Latus Rectum = 2b²/a
The asymptotes are intersecting lines that the hyperbola approaches at infinity: for horizontal orientation, slopes = ±(b/a); for vertical, slopes = ±(a/b). The eccentricity of any hyperbola is always greater than 1, distinguishing it from ellipses (e<1).
Given center (h,k), semi-transverse a, semi-conjugate b, and orientation, we compute c = √(a²+b²). For horizontal: foci = (h ± c, k), vertices = (h ± a, k), asymptote equations: y - k = ±(b/a)(x - h). For vertical: foci = (h, k ± c), vertices = (h, k ± a), asymptotes: y - k = ±(a/b)(x - h). Eccentricity e = c/a, and latus rectum length = 2b²/a. The interactive graph uses parametric sampling: For horizontal, we draw branches via x = h ± a·cosh(t), y = k + b·sinh(t) for t ∈ [-2.4, 2.4]; for vertical, y = k ± a·cosh(t), x = h + b·sinh(t). This produces a smooth, mathematically accurate curve. Asymptotes are drawn as extended dashed lines crossing the center.
Large radio telescopes such as the Arecibo Observatory employ a secondary reflector shaped as a hyperboloid. The primary parabolic reflector focuses incoming waves toward a hyperbola's focus, which then redirects signals to a receiver at the other focus. Engineers use precise hyperbola parameters to minimize signal loss. Using this calculator, you can model the required a, b, and c values to position the foci accurately — critical for gain and directivity.
| Orientation | Center | a | b | c (focal distance) | Eccentricity (e) | Vertices |
|---|---|---|---|---|---|---|
| Horizontal | (0,0) | 3 | 2 | 3.6056 | 1.202 | (±3,0) |
| Vertical | (0,0) | 2 | 1.5 | 2.5 | 1.25 | (0,±2) |
| Horizontal | (2,-1) | 4 | 3 | 5 | 1.25 | (6,-1) & (-2,-1) |
| Rectangular | (0,0) | 2 | 2 | 2.828 | 1.414 | (±2,0) |
The hyperbola was first studied by Menaechmus (4th century BCE) and later formalized by Apollonius of Perga in "Conics". The focus-directrix property was developed by Pappus. Modern analysis relies on Cartesian coordinates, with Euler and Lagrange contributing to analytic geometry. This calculator's algorithms adhere to definitions verified through standard geometry textbooks (Larson, "Precalculus"; Weisstein, MathWorld). Academic references include Wolfram MathWorld and the open-source "Conic Sections" literature.