Hyperbola Calculator

About Hyperbola Calculator

The Hyperbola Calculator finds all key properties of any hyperbola: center, vertices, foci, asymptotes, eccentricity, semi-axes, and latus rectum. It supports standard form and general second-degree equations, providing step-by-step solutions and an interactive graph showing both branches, asymptotes, and the auxiliary rectangle.

How to Use the Hyperbola Calculator

1. Choose the equation form: Select Standard Form to enter the semi-axes (a, b) and center (h, k) directly, or General Form (\(Ax^2 + Cy^2 + Dx + Ey + F = 0\)) for the general equation.
2. Select the orientation (standard form only): Choose whether the transverse axis is horizontal or vertical.
3. Enter the values: Fill in the coefficients or parameters. Use the quick examples to try preset hyperbolas instantly.
4. Click "Calculate Hyperbola" to compute all properties including vertices, foci, asymptotes, eccentricity, and more.
5. Explore the interactive graph: View the color-coded diagram showing both branches, center, vertices, foci, asymptotes, and the auxiliary rectangle.

What Is a Hyperbola?

A hyperbola is a type of conic section formed when a plane intersects both nappes (halves) of a double cone. It consists of two separate open curves called branches. Formally, a hyperbola is the set of all points in a plane where the absolute difference of the distances to two fixed points (the foci) is constant and equal to \(2a\).

Standard Forms of the Hyperbola Equation

There are two standard forms depending on the orientation of the transverse axis:

• Horizontal transverse axis: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) — The hyperbola opens left and right, with vertices at \((h \pm a,\ k)\).
• Vertical transverse axis: \(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\) — The hyperbola opens up and down, with vertices at \((h,\ k \pm a)\).

Here \((h, k)\) is the center, \(a\) is the semi-transverse axis, and \(b\) is the semi-conjugate axis.

Key Components of a Hyperbola

• Center: The midpoint between the two vertices, located at \((h, k)\).
• Vertices: The two points on the hyperbola closest to the center, at distance \(a\) from the center along the transverse axis.
• Foci: Two fixed points at distance \(c = \sqrt{a^2 + b^2}\) from the center. The defining property of a hyperbola involves these points.
• Asymptotes: Two lines through the center that the branches approach but never touch. For a horizontal hyperbola: \(y - k = \pm \frac{b}{a}(x - h)\).
• Eccentricity: \(e = \frac{c}{a}\), always greater than 1. Measures how "open" the branches are — higher values mean flatter, more open branches.
• Latus Rectum: A chord through each focus perpendicular to the transverse axis, with length \(\frac{2b^2}{a}\).
• Conjugate Axis: The axis perpendicular to the transverse axis, with length \(2b\). Together with the transverse axis, it defines the auxiliary rectangle.

Hyperbola vs. Ellipse

While both are conic sections, they differ fundamentally:

• A hyperbola uses the difference of distances to foci; an ellipse uses the sum.
• For a hyperbola, \(c^2 = a^2 + b^2\); for an ellipse, \(c^2 = a^2 - b^2\).
• Hyperbola eccentricity \(e > 1\); ellipse eccentricity \(0 < e < 1\).
• A hyperbola has two separate branches; an ellipse is a single closed curve.

Real-World Applications

• Navigation (LORAN): Uses hyperbolic curves from time-difference-of-arrival signals to determine positions at sea.
• Astronomy: Some comets follow hyperbolic orbits around the Sun, passing through once without returning.
• Cooling towers: The distinctive shape of nuclear power plant cooling towers is a hyperboloid of revolution, which provides structural strength with minimal material.
• Sonic booms: The shock wave from supersonic aircraft forms a hyperbolic intersection with the ground.
• Optics: Hyperbolic mirrors are used in telescope designs (Cassegrain reflectors) to redirect light to a convenient focal point.

FAQ

What is a hyperbola?

A hyperbola is a conic section formed by the set of all points where the absolute difference of distances to two fixed points (foci) is constant. It consists of two separate branches that open in opposite directions and approach but never touch two diagonal lines called asymptotes.

How do you find the foci of a hyperbola?

For a hyperbola in standard form, compute c = sqrt(a² + b²). For a horizontal hyperbola centered at (h, k), the foci are at (h ± c, k). For a vertical hyperbola, the foci are at (h, k ± c).

What are the asymptotes of a hyperbola?

The asymptotes are two straight lines that the hyperbola approaches but never crosses. For a horizontal hyperbola, they are y - k = ±(b/a)(x - h). For a vertical hyperbola, they are y - k = ±(a/b)(x - h).

What is the eccentricity of a hyperbola?

The eccentricity of a hyperbola is e = c/a, where c is the focal distance and a is the semi-transverse axis. For all hyperbolas, e is always greater than 1. A larger eccentricity means the branches are more open and flatter.

What is the difference between a hyperbola and an ellipse?

Both are conic sections, but a hyperbola has two separate branches while an ellipse is a closed curve. For a hyperbola c² = a² + b² and eccentricity is greater than 1, while for an ellipse c² = a² - b² and eccentricity is less than 1. Also, the definition uses the difference of distances for hyperbolas versus the sum for ellipses.

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