Find Focus of Ellipse Calculator

Compute ellipse foci from center, axes, and orientation. Visualize the curve with plotted focal points. Export tables instantly and verify every intermediate value clearly.

Calculator Inputs

Center x-coordinate (h)

Center y-coordinate (k)

Semi-major axis (a)

Semi-minor axis (b)

Major axis orientation

Decimal precision

Reset

Use a ≥ b. When a = b, the figure becomes a circle and both foci merge at the center.

Example Data Table

#	Center	a	b	Orientation	c = √(a² - b²)	Focus 1	Focus 2
1	(0, 0)	5	3	Horizontal	4	(-4, 0)	(4, 0)
2	(2, -1)	6	4	Horizontal	4.472	(-2.472, -1)	(6.472, -1)
3	(-3, 1)	7	2	Vertical	6.708	(-3, -5.708)	(-3, 7.708)
4	(4, 3)	8	5	Vertical	6.245	(4, -3.245)	(4, 9.245)

Formula Used

Horizontal major axis

Standard form: ((x - h)² / a²) + ((y - k)² / b²) = 1

Foci: (h ± c, k), where c = √(a² - b²)

Vertical major axis

Standard form: ((x - h)² / b²) + ((y - k)² / a²) = 1

Foci: (h, k ± c), where c = √(a² - b²)

Extra geometry values

Eccentricity: e = c / a

Major axis length: 2a

Minor axis length: 2b

Area: πab

How to Use This Calculator

Enter the ellipse center coordinates h and k.
Type the semi-major axis value a.
Type the semi-minor axis value b.
Select whether the major axis is horizontal or vertical.
Choose the decimal precision for displayed values.
Click Find Foci to calculate both focal points.
Review the equation, vertices, eccentricity, and plotted ellipse.
Use the export buttons to save results as CSV or PDF.

Frequently Asked Questions

1. What does this calculator find?

It finds both foci of an axis-aligned ellipse. It also returns the center, vertices, co-vertices, eccentricity, standard equation, directrices, area, and a plotted graph.

2. What is the meaning of a and b?

The value a is the semi-major axis. The value b is the semi-minor axis. For this calculator, a must be greater than or equal to b.

3. How is c related to the foci?

The value c is the center-to-focus distance. It is computed using c = √(a² - b²). Once c is known, the foci are placed along the major axis.

4. What happens when a equals b?

Then the ellipse becomes a circle. In that special case, c becomes zero and both foci coincide at the center point.

5. Does this handle rotated ellipses?

No. This version handles standard ellipses aligned with the x-axis or y-axis. Rotated ellipses require an additional rotation term and a different coordinate transformation.

6. Why do I need the orientation option?

Orientation tells the calculator whether the major axis runs left-right or up-down. That choice decides whether the focal shift is applied to x or y coordinates.

7. Can I export my results?

Yes. After calculation, you can download a CSV summary or a PDF report that includes the computed values and the example table.

8. What inputs usually cause errors?

The most common issue is entering a semi-minor axis larger than the semi-major axis. Negative or zero axis lengths also fail validation.