Enter Hyperbola Values
Formula Used
Horizontal form: (x - h)² / a² - (y - k)² / b² = 1
Vertical form: (y - k)² / a² - (x - h)² / b² = 1
Focal distance: c = √(a² + b²)
Eccentricity: e = c / a
Latus rectum: L = 2b² / a
Horizontal vertices: (h ± a, k)
Horizontal foci: (h ± c, k)
Vertical vertices: (h, k ± a)
Vertical foci: (h, k ± c)
How to Use This Calculator
- Enter the center values h and k.
- Choose horizontal or vertical orientation.
- Enter the semi-transverse axis value a.
- Enter the semi-conjugate axis value b.
- Select the decimal precision for rounded output.
- Adjust the graph range if you need a wider view.
- Press the calculate button.
- Review the foci, vertices, asymptotes, equation, and graph.
- Use CSV or PDF export for saved results.
Example Data Table
| Case | Center | a | b | Orientation | c | Vertices | Foci |
|---|---|---|---|---|---|---|---|
| Basic horizontal | (0, 0) | 4 | 3 | Horizontal | 5 | (-4, 0), (4, 0) | (-5, 0), (5, 0) |
| Shifted horizontal | (2, -1) | 6 | 2 | Horizontal | 6.3249 | (-4, -1), (8, -1) | (-4.3249, -1), (8.3249, -1) |
| Basic vertical | (1, 2) | 5 | 4 | Vertical | 6.4031 | (1, -3), (1, 7) | (1, -4.4031), (1, 8.4031) |
Understanding Foci and Vertices
A hyperbola is a conic section with two separate branches. Each branch opens away from the center. The calculator focuses on the points that define that shape. The center is written as h, k. The transverse semi-axis is a. The conjugate semi-axis is b. For every standard hyperbola, the focal distance c is found with c squared equals a squared plus b squared.
Why These Points Matter
Vertices sit on the transverse axis. They mark the closest points of each branch to the center. Foci sit farther out on the same axis. The distance relationship between any point on the curve and both foci stays constant. That constant difference equals 2a. This makes foci useful in geometry, optics, navigation, and curve modeling.
Horizontal and Vertical Forms
A horizontal hyperbola opens left and right. Its equation starts with the x term. A vertical hyperbola opens up and down. Its equation starts with the y term. The calculator changes formulas automatically when you choose orientation. It then returns vertices, foci, asymptotes, eccentricity, latus rectum, and graph data.
Using Results Correctly
The value of a must be positive. The value of b must also be positive. Larger a values move vertices farther from the center. Larger b values make asymptotes steeper or flatter, depending on orientation. Eccentricity is always greater than one for a real hyperbola. A larger eccentricity usually means a more open curve.
Graph and Export Benefits
The plot helps you check the branch direction quickly. It also shows the center, foci, vertices, and asymptotes together. This is useful when comparing homework answers or preparing study notes. The CSV file stores exact coordinates in rows. The PDF option creates a simple report for printing. Both exports can support class records, tutoring sessions, and project documentation.
Common Checks
Always confirm units before recording coordinates. Mixed units can make the graph misleading. Round final answers only after calculation. Keep extra decimals for reports. Compare each point with the selected orientation. For horizontal form, foci and vertices change x values. For vertical form, they change y values. This simple check prevents many sign mistakes and common placement errors.
FAQs
1. What are vertices of a hyperbola?
Vertices are the closest points of each branch to the center. They lie on the transverse axis. Their location depends on the orientation and the value of a.
2. What are foci of a hyperbola?
Foci are two fixed points inside the opening direction. They sit farther from the center than the vertices. They help define the curve by a constant distance difference.
3. How is c calculated?
For a standard hyperbola, c is calculated with c = √(a² + b²). This differs from an ellipse, where the focal relation uses subtraction.
4. What does eccentricity mean?
Eccentricity measures how open the hyperbola is. It equals c divided by a. A real hyperbola always has eccentricity greater than one.
5. When is the hyperbola horizontal?
A hyperbola is horizontal when the x term is positive first. Its branches open left and right. Its vertices and foci change along the x-axis.
6. When is the hyperbola vertical?
A hyperbola is vertical when the y term is positive first. Its branches open upward and downward. Its vertices and foci change along the y-axis.
7. What are asymptotes used for?
Asymptotes guide the shape of each branch. The curve gets closer to them but does not cross them in standard graph behavior.
8. Can I export the results?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button for a simple printable report with key results.