Hyperbola Calculator
Build the equation of a hyperbola (centered or shifted) and instantly see center, vertices, foci, asymptotes, and eccentricity.
How to use
Choose the hyperbola type, set the semi-axes a and b, and optionally shift the center with h and k. Click Calculate or let the inputs trigger the result update to see the derived parameters.
Methodology
From a and b we compute c² = a² + b² and eccentricity e = c/a. The displayed equation, vertices, foci, and asymptotes follow the orientation (horizontal or vertical) and include shifts when h and k are non-zero.
Full original guide (expanded)
Reference table
The calculator converts between the standard hyperbola forms, showing the asymptotes and foci for both horizontal and vertical orientations with optional center shifts.
| Form | Asymptotes | Foci |
|---|---|---|
| \(\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = 1\) | \(y - k = \pm \dfrac{b}{a}(x - h)\) | \((h \pm c, k)\), \(c = \sqrt{a^2+b^2}\) |
| \(\dfrac{(y-k)^2}{a^2} - \dfrac{(x-h)^2}{b^2} = 1\) | \(y - k = \pm \dfrac{a}{b}(x - h)\) | \((h, k \pm c)\) |
How this hyperbola calculator works
The tool follows analytic geometry formulas for hyperbolas adapted from standard references and simplifies results into point-slope formats for quick graphing.
Glossary of Terms
- a: Semi-transverse axis (distance from center to vertex along the major axis).
- b: Semi-conjugate axis.
- c: Distance from center to focus, \(c = \sqrt{a^2+b^2}\).
- e: Eccentricity, \(e = c/a\).
Frequently Asked Questions
How do you find the equation of a hyperbola? Use the centered or shifted form with the chosen orientation and plug in a, b, h, and k.
How are foci computed? They lie at distance c from the center along the transverse axis, with \(c² = a² + b²\).
What are the asymptotes? The asymptotes are lines through the center with slopes ±b/a or ±a/b depending on orientation; they are shown in point-slope form for horizontal and vertical cases.