Hyperbola Calculator (foci, eccentricity, asymptotes)

Understanding the hyperbola calculator — concept and principle

A hyperbola is a curve with two separate branches, defined as the set of points whose distances to two fixed points (the foci) differ by a constant. This calculator works with the standard horizontal hyperbola x²/a² − y²/b² = 1, centred at the origin, and gives the foci, the eccentricity, the asymptotes and the length of the latus rectum. Hyperbolas appear in analytic geometry, in the paths of objects that pass a planet or star without being captured, in radio navigation and in cooling-tower shapes.

How it is calculated

• Focal distance: c = √(a² + b²); the foci are at (±c, 0).
• Eccentricity: e = c ÷ a, always greater than 1.
• Asymptotes: y = ±(b ÷ a)x.
• Latus rectum: 2b² ÷ a, the length of the chord through a focus perpendicular to the transverse axis.

The vertices, where the branches turn, are at (±a, 0).

The principle

For any point on the hyperbola, the difference of its distances to the two foci is 2a. Writing that condition with foci at (±c, 0) and simplifying gives x²/a² − y²/(c² − a²) = 1, so b² = c² − a², or c² = a² + b². Unlike an ellipse, where c² = a² − b², the foci lie outside the vertices. For large x, the "1" on the right becomes negligible compared with x²/a² and y²/b², so the curve approaches the lines y = ±(b/a)x; these are the asymptotes, which the branches approach but never touch. Eccentricity measures how open the branches are: e close to 1 gives narrow branches, large e gives branches that open wide.

Worked example

1. Inputs: a = 3, b = 4.
2. c = √(9 + 16) = 5, so the foci are (±5, 0).
3. Eccentricity: 5 ÷ 3 ≈ 1.666666667.
4. Asymptotes: b ÷ a = 4 ÷ 3, so y = ±1.333333333x.
5. Latus rectum: 2 × 16 ÷ 3 = 32 ÷ 3 ≈ 10.66666667.

How to read the result

The vertices (±3, 0) are where the branches are closest to the centre, and the foci (±5, 0) sit further out. The asymptote slope 4/3 shows how steeply the branches climb; when a = b the asymptotes are at 45° and the hyperbola is called rectangular, with e = √2 ≈ 1.414. The eccentricity of conics follows a clear scale: 0 for a circle, between 0 and 1 for an ellipse, exactly 1 for a parabola and above 1 for a hyperbola.

Common mistakes

• Using c² = a² − b² from the ellipse; for a hyperbola, c² = a² + b².
• Swapping a and b. In this form a belongs to the positive x² term, regardless of which number is larger.
• Applying the results to y²/a² − x²/b² = 1. That hyperbola opens up and down, with foci (0, ±c) and asymptotes y = ±(a/b)x.

Frequently asked questions

What are the vertices of the example hyperbola?

(±3, 0). They lie at distance a from the centre, between the centre and the foci.

What is the difference of distances from a point on it to the foci?

Always 2a, here 6. For the vertex (3, 0): 8 − 2 = 6.

Is the latus rectum the same for ellipses?

The formula 2b²/a is the same, but b is defined differently for each curve.

Sources: Apollonius of Perga, Conics, Books I–III · J. Stewart, L. Redlin and S. Watson, Precalculus: Mathematics for Calculus, Cengage, chapter on conic sections