Calculator Inputs
Example Data Table
| Case | Center | Orientation | a | b | Standard form |
|---|---|---|---|---|---|
| Example 1 | (0, 0) | Horizontal | 5 | 3 | x²/25 - y²/9 = 1 |
| Example 2 | (2, -1) | Vertical | 4 | 6 | (y + 1)²/16 - (x - 2)²/36 = 1 |
| Example 3 | (-3, 2) | Horizontal | 7 | 2 | (x + 3)²/49 - (y - 2)²/4 = 1 |
Formula Used
For a horizontal transverse axis, the standard form is (x − h)²/a² − (y − k)²/b² = 1.
For a vertical transverse axis, the standard form is (y − k)²/a² − (x − h)²/b² = 1.
The focal relationship is c² = a² + b². Eccentricity is e = c/a.
The asymptotes are y − k = ±(b/a)(x − h) for horizontal hyperbolas and x − h = ±(b/a)(y − k) for vertical hyperbolas.
The directrices are x = h ± a/e for horizontal cases and y = k ± a/e for vertical cases.
The latus rectum length is 2b²/a.
How to Use This Calculator
- Select the input mode that matches your known values.
- Choose whether the transverse axis is horizontal or vertical.
- Enter the center values. Add either a and b, or a vertex and focus.
- Press Calculate Hyperbola to show results above the form.
- Review the standard form, general form, geometry points, and asymptotes.
- Use the CSV or PDF buttons to save the result set.
Frequently Asked Questions
1. What does this calculator solve?
It builds a hyperbola from either standard parameters or from a center, one vertex, and one focus. It also returns important derived properties.
2. What is the difference between a and b?
a is the semi-transverse axis that reaches each vertex. b is the semi-conjugate axis that controls the asymptote slope and branch spread.
3. Why must c be greater than a?
For a real hyperbola, each focus lies farther from the center than each vertex. That requirement makes b² = c² − a² positive.
4. Can this calculator show the expanded equation?
Yes. It provides the standard form and an expanded quadratic form, which is useful for algebra checks, assignments, and graphing software.
5. What happens if I enter a vertical hyperbola?
The calculator switches to the vertical standard template, updates the asymptotes, and places the vertices and foci along the y-direction.
6. What are conjugate endpoints?
They are the endpoints of the conjugate axis through the center. They are not on the hyperbola, but they help describe its geometry.
7. Does the tool support decimal values?
Yes. All coordinate and axis inputs accept decimal values, making the calculator suitable for classroom work, engineering contexts, and verification tasks.
8. How do the export buttons work?
The CSV button downloads the displayed results as a spreadsheet-friendly file. The PDF button opens a print-ready version that you can save.