Hyperbola Calculator — Vertices, Foci & Asymptotes
Calculate all properties of a hyperbola from its equation: vertices, co-vertices, foci, asymptotes, eccentricity, and directrix.
Hyperbola Guide
Hyperbola Properties
Horizontal hyperbola x²/a² − y²/b² = 1: opens left and right. Vertices: (±a, 0). Foci: (±c, 0) where c = √(a² + b²). Asymptotes: y = ±(b/a)x. Vertical hyperbola y²/a² − x²/b² = 1: opens up and down. Vertices: (0, ±a). Foci: (0, ±c). Asymptotes: y = ±(a/b)x. Key identity (different from ellipse): c² = a² + b² (ellipse: c² = a² − b²). For hyperbola, c > a always. Eccentricity: e = c/a > 1 always (vs ellipse: 0 < e < 1, circle: e = 0).
Eccentricity
Eccentricity e = c/a measures how stretched a conic section is. For a hyperbola: e > 1 always. Close to 1: narrow hyperbola (almost like two lines). Large e: very wide, flat hyperbola. Rectangular hyperbola (e = √2): asymptotes are perpendicular (y = ±x for a = b). xy = k is also a rectangular hyperbola (rotated 45°). Applications: Rutherford scattering — alpha particles fired at a gold foil follow hyperbolic paths due to Coulomb repulsion. LORAN navigation used hyperbolic position lines. Conic
Asymptotes
Asymptotes are the lines the hyperbola approaches but never touches. For x²/a² − y²/b² = 1: asymptotes y = ±(b/a)x. For y²/a² − x²/b² = 1: asymptotes y = ±(a/b)x. The rectangle formed by the vertices and co-vertices (b, 0) has the asymptotes as diagonals — this is the easiest way to draw asymptotes by hand. A hyperbola with shifted centre (h, k): asymptotes become y − k = ±(b/a)(x − h). The asymptotes and the latus rectum (the chord through a focus perpendicular to the transverse axis) are key f
Relationship to Other Conics
All conic sections can be defined as the set of points where the ratio of distance to focus to distance to directrix = eccentricity (e). Circle: e = 0 (no directrix — centre is equidistant from all points). Ellipse: 0 < e < 1 (directrix outside the curve). Parabola: e = 1 (directrix is perpendicular to axis, parallel to the opening). Hyperbola: e > 1 (directrix between the vertices). All four conics arise from intersecting a cone with a plane at different angles. This unified definition (attribu
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