Calculator Inputs
Formula Used
Standard ellipse: ((x - h)² / a²) + ((y - k)² / b²) = 1
Focal distance: c = √(a² - b²)
Horizontal foci: (h ± c, k)
Vertical foci: (h, k ± c)
Eccentricity: e = c / a
Area: Area = πab
Latus rectum: L = 2b² / a
How to Use This Calculator
- Choose standard form or general equation form.
- Enter the center and semi-axis values for standard form.
- Use a larger value for the semi-major axis.
- Enter an angle if the major axis is tilted.
- For general form, enter the five equation coefficients.
- Press the calculate button.
- Read the foci, vertices, eccentricity, area, and graph.
- Use CSV or PDF export for records.
Example Data Table
| Center | Semi-major a | Semi-minor b | Angle | c value | Foci |
|---|---|---|---|---|---|
| (0, 0) | 5 | 3 | 0° | 4 | (4, 0), (-4, 0) |
| (2, -1) | 10 | 6 | 90° | 8 | (2, 7), (2, -9) |
| (1, 3) | 8 | 5 | 30° | 6.245 | Rotated around center |
Understanding Ellipse Foci
The role of foci
An ellipse is shaped around two fixed points. These points are called foci. Every point on the ellipse has a special property. The total distance from that point to both foci stays constant. This constant total equals the full major axis length. This idea makes foci useful in geometry, optics, astronomy, design, and engineering.
Why the major axis matters
The foci always lie on the major axis. The major axis is the longest direction across the ellipse. If the ellipse is wide, the foci move left and right from the center. If the ellipse is tall, the foci move up and down. If an angle is used, the foci rotate with the major axis.
How distance c is found
The calculator first identifies the semi-major axis a and semi-minor axis b. It then applies c equals the square root of a squared minus b squared. A circle has equal axes, so c becomes zero. In that case, both foci meet at the center. A long narrow ellipse has a larger c value.
Using equation form
Many problems give an equation instead of center and axis values. This tool can handle aligned general equations. It completes the square, finds the center, and converts the expression into standard form. Then it compares the denominators. The larger denominator gives the major direction.
Reading the results
The focus coordinates show the two key internal points. Vertices show the endpoints of the major axis. Co-vertices show the endpoints of the minor axis. Eccentricity describes how stretched the ellipse is. Area gives the enclosed space. The graph helps verify the position visually.
Practical use
Students can check homework steps. Teachers can prepare examples. Designers can estimate curved layouts. Analysts can compare shapes. The export options make it easy to save each calculation. Use consistent units for every length. The same unit will apply to the foci, axes, latus rectum, and area output.
FAQs
1. What are the foci of an ellipse?
The foci are two fixed points inside an ellipse. The sum of distances from any ellipse point to both foci stays constant.
2. What formula finds the foci?
Use c = √(a² - b²). Then place c units from the center along the major axis.
3. Which axis contains the foci?
The foci always lie on the major axis. This is the longer axis of the ellipse.
4. What happens when a equals b?
The ellipse becomes a circle. The value of c is zero, so both foci meet at the center.
5. Can this calculator handle vertical ellipses?
Yes. Use a 90 degree angle in standard mode, or enter a general equation with the larger denominator under y.
6. Can I use negative center coordinates?
Yes. The center values h and k can be positive, negative, or zero.
7. What is eccentricity?
Eccentricity measures how stretched an ellipse is. It equals c divided by a and stays between 0 and 1.
8. Does the export include all results?
Yes. The CSV and PDF exports include foci, axes, center, eccentricity, area, vertices, and latus rectum values.