Ellipse Focus Calculator
Enter center coordinates, semi-axis lengths, and orientation. The result appears above this form after submission.
Example Data Table
| Case | Center (h, k) | a | b | Orientation | c = √(a² - b²) | Foci |
|---|---|---|---|---|---|---|
| Example 1 | (0, 0) | 6 | 4 | Horizontal | 4.4721 | (-4.4721, 0), (4.4721, 0) |
| Example 2 | (2, -1) | 7 | 3 | Vertical | 6.3246 | (2, -7.3246), (2, 5.3246) |
| Example 3 | (-3, 5) | 5 | 2 | Horizontal | 4.5826 | (-7.5826, 5), (1.5826, 5) |
Formula Used
For an ellipse, the semi-major axis is a and the semi-minor axis is b. The focal distance from the center is c.
Core relation: c = √(a² − b²)
Eccentricity: e = c / a
Focal length: 2c
Latus rectum length: 2b² / a
Area: πab
Perimeter approximation: π[3(a + b) − √((3a + b)(a + 3b))]
Standard Forms
Horizontal major axis: ((x − h)² / a²) + ((y − k)² / b²) = 1
Foci: (h − c, k) and (h + c, k)
Vertical major axis: ((x − h)² / b²) + ((y − k)² / a²) = 1
Foci: (h, k − c) and (h, k + c)
How to Use This Calculator
- Enter the ellipse center coordinates h and k.
- Select whether the major axis is horizontal or vertical.
- Enter semi-major axis a.
- Enter semi-minor axis b.
- Make sure a is greater than or equal to b.
- Click Calculate Focus.
- Read the focus coordinates, eccentricity, vertices, and directrices.
- Use the graph to inspect the ellipse visually.
- Download the computed results as CSV or PDF.
Frequently Asked Questions
1. What is the focus of an ellipse?
An ellipse has two fixed points called foci. For any point on the ellipse, the sum of distances to both foci stays constant. These points define the ellipse shape and eccentricity.
2. How do I calculate the foci?
First compute c = √(a² − b²). For a horizontal ellipse, the foci are (h ± c, k). For a vertical ellipse, the foci are (h, k ± c).
3. Why must a be greater than or equal to b?
The semi-major axis is the longest radius of the ellipse. That value is always a. If b exceeds a, then the labels are reversed and the input no longer matches the standard ellipse definition.
4. What happens when a equals b?
When a equals b, the ellipse becomes a circle. Then c becomes zero, both foci merge at the center, and eccentricity becomes zero.
5. What does eccentricity show?
Eccentricity measures how stretched the ellipse is. Values near zero look more circular. Values closer to one look more elongated.
6. Does orientation change the focal formula?
Yes. Horizontal ellipses shift the foci left and right from the center. Vertical ellipses shift the foci up and down from the center.
7. What is the latus rectum?
The latus rectum is a chord through a focus and perpendicular to the major axis. Its length for an ellipse is 2b² / a.
8. Can I use decimal inputs?
Yes. The calculator accepts integers and decimals for center coordinates and axis lengths. That makes it useful for classroom work, design tasks, and analytic geometry problems.