Enter Ellipse Values
Formula Used
Standard ellipse form:
((x - h)² / a²) + ((y - k)² / b²) = 1
General conic form:
Ax² + Bxy + Cy² + Dx + Ey + F = 0
For an axis-aligned ellipse:
A = b²
B = 0
C = a²
D = -2hb²
E = -2ka²
F = b²h² + a²k² - a²b²
How to Use This Calculator
- Enter the center value h from the expression x - h.
- Enter the center value k from the expression y - k.
- Enter the x semi-axis length. This is a positive value.
- Enter the y semi-axis length. This is also positive.
- Select the decimal precision and coefficient mode.
- Press the calculate button to expand the ellipse.
- Review the general form, coefficients, graph, and steps.
- Use the CSV or PDF buttons to save the result.
Example Data Table
| h | k | x Semi-Axis | y Semi-Axis | Standard Form | General Form |
|---|---|---|---|---|---|
| 2 | -1 | 5 | 3 | ((x - 2)² / 25) + ((y + 1)² / 9) = 1 | 9x² + 25y² - 36x + 50y - 164 = 0 |
| 0 | 3 | 4 | 2 | (x² / 16) + ((y - 3)² / 4) = 1 | 4x² + 16y² - 96y + 80 = 0 |
| -2 | 1 | 6 | 3 | ((x + 2)² / 36) + ((y - 1)² / 9) = 1 | 9x² + 36y² + 36x - 72y - 288 = 0 |
Ellipse Equation Conversion Guide
What This Calculator Does
An ellipse often appears first in standard form. That form is useful because it shows the center and semi-axis lengths quickly. General form is different. It writes every term on one side of the equation. This calculator converts the standard form into that expanded conic form. It also shows the key coefficients used in many algebra and analytic geometry problems.
Why General Form Matters
General form helps when comparing conic sections. It is also helpful when solving systems, checking a graph, or using symbolic algebra. The coefficients A, B, C, D, E, and F describe the equation structure. For a non-rotated ellipse, the xy term is zero. That means B equals zero. The x squared and y squared coefficients stay positive after normal expansion.
Understanding the Inputs
The center values are h and k. They move the ellipse left, right, up, or down. The x semi-axis controls horizontal spread. The y semi-axis controls vertical spread. These values must be positive. When the x semi-axis is larger, the major axis is horizontal. When the y semi-axis is larger, the major axis is vertical.
How the Expansion Works
The calculator first squares both semi-axis values. Then it multiplies the full equation by the product of those squared values. This removes the fractions. Next, it expands each binomial. Finally, it collects all terms on the left side. The result becomes Ax² + Bxy + Cy² + Dx + Ey + F = 0.
Using the Results
The output includes the standard equation, general equation, coefficients, vertices, foci, area, eccentricity, and latus rectum. The graph gives a quick visual check. The CSV file is useful for spreadsheets. The PDF file is useful for notes, assignments, and records. Always check whether your input uses semi-axis lengths, not squared denominators.
FAQs
1. What is ellipse general form?
Ellipse general form is Ax² + Bxy + Cy² + Dx + Ey + F = 0. For an axis-aligned ellipse, B is zero. This calculator expands standard form into that format.
2. What values should I enter for a and b?
Enter the semi-axis lengths. Do not enter the squared denominators unless they are already square roots. For example, if the denominator is 25, the semi-axis is 5.
3. Can h or k be negative?
Yes. Negative center values are allowed. The calculator handles signs automatically and writes the correct expanded linear terms in the general equation.
4. Why is the xy coefficient zero?
This calculator handles non-rotated ellipses. A rotated ellipse has an xy term. Standard axis-aligned ellipse equations do not include rotation, so B equals zero.
5. What is compact integer form?
Compact integer form divides whole-number coefficients by their greatest common divisor when possible. It keeps the same ellipse but displays a cleaner equation.
6. Does the calculator show graph points?
Yes. The Plotly graph draws the ellipse from calculated parametric points. It helps you check the center, width, height, and orientation visually.
7. What is eccentricity?
Eccentricity measures how stretched the ellipse is. A value near zero means the ellipse is close to a circle. A larger value means more stretching.
8. Can I download the result?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button to save a readable report with the main equation and values.