Enter Ellipse Values
Use center coordinates, axis lengths, and orientation to convert an ellipse into standard form.
Formula Used
Horizontal ellipse:
((x - h)² / a²) + ((y - k)² / b²) = 1
Vertical ellipse:
((x - h)² / b²) + ((y - k)² / a²) = 1
Focus distance: c = √(a² - b²)
Eccentricity: e = c / a
Area: A = πab
Perimeter estimate: π[3(a+b) - √((3a+b)(a+3b))]
How to Use This Calculator
- Enter the center coordinates h and k.
- Enter the semi-major axis as a.
- Enter the semi-minor axis as b.
- Select horizontal or vertical orientation.
- Choose a graph angle step for coordinate sampling.
- Press the calculate button.
- Review the standard equation, foci, vertices, area, and graph.
- Use CSV or PDF buttons to save the result.
Example Data Table
| Center h | Center k | a | b | Orientation | Standard Form |
|---|---|---|---|---|---|
| 2 | -1 | 6 | 3 | Horizontal | ((x - 2)² / 36) + ((y + 1)² / 9) = 1 |
| 0 | 4 | 5 | 2 | Vertical | (x² / 4) + ((y - 4)² / 25) = 1 |
| -3 | 1 | 8 | 4 | Horizontal | ((x + 3)² / 64) + ((y - 1)² / 16) = 1 |
Understanding the Standard Form of an Ellipse
What the Equation Shows
The standard form of an ellipse shows its center, axis lengths, and direction. It also gives a clear path to find vertices and foci. This calculator builds that equation from simple inputs. You only need the center and two semi-axis values. The larger value is the semi-major axis. The smaller value is the semi-minor axis.
Why the Center Matters
The center is written as h and k. It moves the ellipse left, right, up, or down. When h changes, the equation shifts along the x-axis. When k changes, the equation shifts along the y-axis. This makes the equation useful for plotted data and statistical shapes.
Major and Minor Axes
The major axis is the longest width of the ellipse. The minor axis is the shorter width. A horizontal ellipse places a² below the x expression. A vertical ellipse places a² below the y expression. This choice controls the position of vertices and foci.
Foci and Eccentricity
The foci show how stretched the ellipse is. They are placed inside the ellipse on the major axis. The value c measures the distance from the center to each focus. Eccentricity compares c with a. A value near zero looks more circular. A value closer to one looks more stretched.
Practical Use
The calculator helps with analytic geometry, statistics diagrams, and curve modeling. It also supports reports by giving downloadable results. The graph makes the shape easier to verify. The coordinate table helps users inspect sampled points. These features make the tool useful for students, teachers, analysts, and developers.
FAQs
1. What is the standard form of an ellipse?
It is a clean equation that shows the center, axis lengths, and orientation of an ellipse. It usually equals one.
2. What do h and k mean?
They represent the center of the ellipse. The point is written as (h, k) on the coordinate plane.
3. What is the semi-major axis?
The semi-major axis is half of the longest diameter. It is marked as a in the standard equation.
4. What is the semi-minor axis?
The semi-minor axis is half of the shortest diameter. It is marked as b in the standard equation.
5. How are foci calculated?
Foci use the formula c = √(a² - b²). They sit along the major axis inside the ellipse.
6. What does eccentricity mean?
Eccentricity measures how stretched the ellipse is. A smaller value means the shape is closer to a circle.
7. Can this calculator graph the ellipse?
Yes. It creates graph points from angle samples and displays the ellipse with center, vertices, and foci.
8. Can I export the result?
Yes. You can download a CSV file or a PDF report after calculating the ellipse values.