Calculator Inputs
Example Data Table
| Conic | Center or Vertex | Orientation | a | b | p | Main Output |
|---|---|---|---|---|---|---|
| Ellipse | (0, 0) | Horizontal | 5 | 3 | Not used | Vertices (-5, 0), (5, 0); foci (-4, 0), (4, 0) |
| Hyperbola | (2, 1) | Vertical | 4 | 2 | Not used | Vertices (2, -3), (2, 5); foci use c = sqrt(20) |
| Parabola | (1, -2) | Horizontal | Not used | Not used | 3 | Vertex (1, -2); focus (4, -2); directrix x = -2 |
Formula Used
Ellipse
For an ellipse, use c = √(a² - b²). The vertices are a units from the center along the major axis. The foci are c units from the center along that same axis. Eccentricity is e = c / a.
Hyperbola
For a hyperbola, use c = √(a² + b²). The vertices are a units from the center along the transverse axis. The foci are c units from the center along that axis. Eccentricity is e = c / a.
Parabola
For a parabola, p is the directed distance from vertex to focus. A horizontal parabola uses (y - k)² = 4p(x - h). A vertical parabola uses (x - h)² = 4p(y - k).
How to Use This Calculator
- Select ellipse, hyperbola, or parabola.
- Choose horizontal or vertical orientation.
- Enter h and k for the center or vertex.
- Enter a and b for ellipse or hyperbola.
- Enter p for a parabola.
- Choose decimal precision.
- Press Calculate to view results.
- Use CSV or PDF buttons to export the same calculation.
Vertices and Foci Calculator Guide
Purpose of the Tool
This calculator finds important points of common conic sections. It works with ellipses, hyperbolas, and parabolas. You can enter a center, axis lengths, orientation, and rounding choice. The tool then returns vertices, foci, directrices, eccentricity, and other helpful values. It is useful for algebra, analytic geometry, graphing, tutoring, and quick homework checks.
Understanding the Inputs
The values h and k locate the center for ellipses and hyperbolas. For parabolas, they locate the vertex. The value a is the semi-major axis for an ellipse. It is the transverse semi-axis for a hyperbola. The value b is the second semi-axis. A parabola does not use a and b. It uses p instead. The sign of p controls the opening direction. Positive p opens right or upward. Negative p opens left or downward.
Why Orientation Matters
Orientation tells the calculator where the main axis lies. A horizontal ellipse places vertices and foci left and right of the center. A vertical ellipse places them above and below. A horizontal hyperbola opens left and right. A vertical hyperbola opens upward and downward. For a parabola, orientation decides whether the squared term is x or y.
Reading the Results
The result table gives the final coordinates first. It also shows the directrix or directrices when they apply. For hyperbolas, it includes asymptotes. For parabolas, it includes latus rectum endpoints. These extra values help you sketch the graph accurately. They also help check whether your equation has been entered in the correct form.
Exporting Work
The CSV export is best for spreadsheets. The PDF export is best for notes, worksheets, or reports. Both options use the same submitted inputs. You can change any value, calculate again, and export a fresh result. This keeps the workflow simple and avoids copying errors during repeated practice.
FAQs
1. What does this calculator find?
It finds vertices, foci, directrices, eccentricity, and related values for ellipses, hyperbolas, and parabolas.
2. What is a vertex?
A vertex is a key turning or boundary point on a conic. Ellipses and hyperbolas have two main vertices. A parabola has one vertex.
3. What is a focus?
A focus is a fixed point used to define a conic. Its position controls the shape, spread, and eccentricity of the curve.
4. What does a mean?
For an ellipse, a is the semi-major axis. For a hyperbola, a is the transverse semi-axis. It must be positive.
5. What does b mean?
The value b is the other semi-axis for ellipses and hyperbolas. It helps calculate the focal distance and graph shape.
6. What does p mean for a parabola?
The value p is the directed distance from the vertex to the focus. Its sign controls the opening direction.
7. Can I export the result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a simple printable result sheet.
8. Why must ellipse a be greater than b?
This version treats a as the semi-major axis. That keeps the focal formula clear and prevents invalid square roots.