Ellipse equation foci calculator
Choose an input mode, define the center, and calculate foci, equation, directrices, and graph details from one page.
Formula used
Core ellipse relationship
c² = a² - b², where a is the semi-major axis, b is the semi-minor axis, and c is the distance from center to each focus.
Standard equations
Horizontal major axis: ((x - h)² / a²) + ((y - k)² / b²) = 1
Vertical major axis: ((x - h)² / b²) + ((y - k)² / a²) = 1
Focus coordinates
Horizontal: (h - c, k) and (h + c, k)
Vertical: (h, k - c) and (h, k + c)
Mode conversions
Given a and b: c = √(a² - b²)
Given a and c: b = √(a² - c²)
Given b and c: a = √(b² + c²)
Extra outputs
Eccentricity: e = c / a
Area: A = πab
Latus rectum: 2b² / a
Directrix distance: a² / c from the center
How to use this calculator
- Select the input mode that matches your known ellipse values.
- Choose whether the major axis is horizontal or vertical.
- Enter the center coordinates h and k for shifted ellipses.
- Fill the two main values and choose display decimals.
- Press calculate to view foci, equation, graph, and export options.
Example data table
| Mode | Orientation | Center | Inputs | a | b | Foci |
|---|---|---|---|---|---|---|
| a and b | Horizontal | (0, 0) | a = 5, b = 3 | 5 | 3 | (-4, 0), (4, 0) |
| a and c | Vertical | (2, -1) | a = 13, c = 5 | 13 | 12 | (2, -6), (2, 4) |
| b and c | Horizontal | (-3, 2) | b = 4, c = 3 | 5 | 4 | (-6, 2), (0, 2) |
FAQs
1. What does the value a represent?
The value a is the semi-major axis. It is half the longest diameter of the ellipse. This value always exceeds b for a true ellipse.
2. Why must a be greater than b?
An ellipse needs a longer major axis than minor axis. When a equals b, the shape becomes a circle, and the foci merge at the center.
3. Can this calculator handle shifted ellipses?
Yes. Enter any center values for h and k. The calculator updates the standard equation, foci, vertices, directrices, and graph around that shifted center.
4. How is the focal distance c calculated?
When a and b are known, the focal distance uses c = √(a² - b²). That distance measures how far each focus sits from the center.
5. What changes when I pick vertical orientation?
The major axis rotates to the y-direction. Foci and vertices move above and below the center, and the equation places a² under the y term.
6. Is the graph drawn with equal axis scaling?
Yes. The chart locks both axes to equal scale. This keeps the ellipse shape accurate and prevents visual stretching during plotting.
7. What do the export buttons download?
The CSV button downloads the result table as spreadsheet-ready text. The PDF button creates a simple report containing the same calculated output table.
8. When should I use the b and c mode?
Use b and c mode when the semi-minor axis and focal distance are known already. The calculator then finds a before plotting all other details.