BYTETOOLS

Hyperbola Calculator

Find the center, vertices, foci, eccentricity, asymptote equations and latus rectum of a hyperbola from standard form or a general second-degree equation.

5
c (focal distance)
1.666667
Eccentricity
1.333333
Asymptote slope
10.6667
Latus rectum

Hyperbola properties

x²/9 - y²/16 = 1

  • Center: (0, 0)
  • Opens: left and right
  • Vertices: (-3, 0) and (3, 0)
  • Co-vertices: (0, -4) and (0, 4)
  • Foci: (-5, 0) and (5, 0)
  • Transverse axis length 2a = 6, conjugate axis length 2b = 8
  • Asymptotes: y = ±1.333333·x
  • Directrices: x = -1.8 and x = 1.8

Sketch

The dashed rectangle is the central box, 2a by 2b. Its diagonals are the asymptotes the branches approach.

How these are derived

For a hyperbola the focal distance adds rather than subtracts: c = √(a² + b²), so the eccentricity c/a is always greater than 1. The asymptotes are the diagonals of the 2a × 2b central box, with slope ±b/a when the hyperbola opens sideways and ±a/b when it opens up and down. The latus rectum, 2b²/a, is the chord through a focus perpendicular to the transverse axis.

What is the Hyperbola Calculator?

The ByteTools Hyperbola Calculator solves both orientations of the standard form — (x - h)²/a² - (y - k)²/b² = 1 and its vertical twin — as well as a general equation Ax² + Cy² + Dx + Ey + F = 0 whose squared terms have opposite signs.

  • Handles both horizontal and vertical hyperbolas, and detects which one a general equation is
  • Center, vertices, co-vertices, foci, directrices and axis lengths in one panel
  • Asymptote equations printed in point-slope form
  • Flags the degenerate two-crossing-lines case and the ellipse case
  • Sketch of both branches with the central box and asymptotes drawn
  • Runs entirely in your browser — nothing leaves your device

How to use the Hyperbola Calculator

  1. 1

    Choose standard form or general form from the first dropdown.

  2. 2

    In standard form, pick whether the hyperbola opens left-right or up-down, then enter a, b and the center.

  3. 3

    In general form, enter the coefficients A, C, D, E and F — A and C must have opposite signs.

  4. 4

    Read c, eccentricity, asymptote slope and latus rectum in the summary tiles.

  5. 5

    Check the full property list and the sketch, then click Copy results.

About the Hyperbola Calculator

The ByteTools Hyperbola Calculator solves both orientations of the standard form — (x - h)²/a² - (y - k)²/b² = 1 and its vertical twin — as well as a general equation Ax² + Cy² + Dx + Ey + F = 0 whose squared terms have opposite signs. It completes the square, works out which way the curve opens, and reports every named feature.

You get the center, vertices, co-vertices, foci from c = √(a² + b²), eccentricity c/a, the two asymptote equations, the directrices, and the latus rectum 2b²/a. A sketch draws both branches through the central 2a by 2b box whose diagonals are the asymptotes, so the geometry is visible rather than just listed.

Everything runs client-side in your browser, so nothing is uploaded and the page works offline. The degenerate case, where the equation collapses to a pair of crossing lines, is detected and explained instead of being silently rounded away.

Frequently asked questions

How do you find the foci of a hyperbola?

Use c = √(a² + b²) — note the plus sign, unlike an ellipse — then move c units from the center along the transverse axis in both directions. For x²/9 - y²/16 = 1, a = 3 and b = 4 give c = 5, so the foci sit at (-5, 0) and (5, 0).

What are the asymptotes of a hyperbola?

They are the two straight lines the branches approach but never touch. For a hyperbola opening left and right they are y - k = ±(b/a)(x - h); for one opening up and down the slopes become ±a/b. They are the diagonals of the 2a by 2b rectangle centered on the hyperbola.

Can the eccentricity of a hyperbola be less than 1?

No. Because c = √(a² + b²) is always larger than a, the eccentricity c/a is always greater than 1. Ellipses have eccentricity between 0 and 1, parabolas exactly 1, and hyperbolas above 1 — that is what separates the three conic sections.

How do I tell a hyperbola from an ellipse in general form?

Compare the signs of the x² and y² coefficients. Opposite signs mean a hyperbola; matching signs mean an ellipse or circle. If one of them is missing entirely the curve is a parabola. This tool checks the signs first and tells you when the equation is not a hyperbola.

What does it mean when the equation gives two crossing lines?

If completing the square leaves zero on the right-hand side, the hyperbola has degenerated into its own asymptotes: a pair of straight lines crossing at the center. It is a real solution set, just not a curve, and the calculator names the crossing point.

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