Hyperbola Calculator Form
Example Data Table
Use these sample values to test the calculator.
| Case | Center | a | b | Orientation | Vertices | Foci |
|---|---|---|---|---|---|---|
| Standard horizontal | (0, 0) | 5 | 3 | Horizontal | (-5, 0), (5, 0) | (-5.831, 0), (5.831, 0) |
| Shifted horizontal | (2, -1) | 4 | 2 | Horizontal | (-2, -1), (6, -1) | (-2.472, -1), (6.472, -1) |
| Vertical hyperbola | (1, 3) | 6 | 4 | Vertical | (1, -3), (1, 9) | (1, -4.211), (1, 10.211) |
Formula Used
Horizontal hyperbola:
((x - h)^2 / a^2) - ((y - k)^2 / b^2) = 1
Vertical hyperbola:
((y - k)^2 / a^2) - ((x - h)^2 / b^2) = 1
The center is (h, k). The semi-transverse axis is a. The semi-conjugate axis is b.
The focal distance is calculated with c = √(a² + b²).
For a horizontal hyperbola, vertices are (h - a, k) and (h + a, k).
For a horizontal hyperbola, foci are (h - c, k) and (h + c, k).
For a vertical hyperbola, vertices are (h, k - a) and (h, k + a).
For a vertical hyperbola, foci are (h, k - c) and (h, k + c).
The eccentricity is e = c / a. The latus rectum length is 2b² / a.
How to Use This Calculator
- Enter the center values h and k.
- Enter the semi-transverse axis value a.
- Enter the semi-conjugate axis value b.
- Select horizontal or vertical orientation.
- Use scale if your input must be multiplied.
- Choose decimal places for cleaner output.
- Press calculate to show results above the form.
- Download CSV or PDF for records.
Hyperbola Vertices and Foci Guide
What the Calculator Finds
A hyperbola is built from a center, two vertices, and two foci. These points describe its shape and direction. This calculator finds those values from standard hyperbola inputs. It also reports asymptotes, eccentricity, and latus rectum length. These extra values help with graphing and checking work.
Why Vertices Matter
Vertices mark the closest points on each branch. They sit on the transverse axis. In a horizontal hyperbola, they move left and right. In a vertical hyperbola, they move up and down. Their distance from the center is always a. This makes them easy to locate once the orientation is known.
Why Foci Matter
Foci control the curve definition. Every hyperbola has two foci. They sit beyond the vertices on the transverse axis. The distance from the center to each focus is c. The value c is not entered directly. It is found from a and b. The calculator uses the relation c squared equals a squared plus b squared.
Horizontal and Vertical Forms
Orientation changes the equation and point locations. A horizontal hyperbola opens left and right. Its x term is positive in the standard equation. A vertical hyperbola opens upward and downward. Its y term is positive in the standard equation. Choosing the correct orientation is important. Wrong orientation gives points on the wrong axis.
Advanced Output Uses
The asymptotes guide the branch direction. They act like boundary lines for sketching. Eccentricity shows how open the hyperbola is. A larger eccentricity means wider separation from the center. The latus rectum length gives another focal measurement. It is useful in analytic geometry. Teachers can use the CSV export for grading examples. Students can save PDF reports for homework notes. Designers can test scaled coordinate systems quickly.
FAQs
1. What are hyperbola vertices?
Vertices are the nearest points of the two branches to the center. They lie on the transverse axis and are a units away from the center.
2. What are hyperbola foci?
Foci are two fixed points that define the hyperbola. They lie on the transverse axis and are c units from the center.
3. How is c calculated?
The calculator uses c = √(a² + b²). This formula works for both horizontal and vertical standard hyperbolas.
4. What does a mean?
The value a is the semi-transverse axis. It measures the distance from the center to each vertex.
5. What does b mean?
The value b is the semi-conjugate axis. It helps calculate foci, asymptote slope, and latus rectum length.
6. When should I choose horizontal orientation?
Choose horizontal orientation when the hyperbola opens left and right. In standard form, the x squared term is positive.
7. When should I choose vertical orientation?
Choose vertical orientation when the hyperbola opens upward and downward. In standard form, the y squared term is positive.
8. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button to save a neat printable report.