Enter center, focus, and vertex to solve precisely. Get axes, eccentricity, directrices, and standard forms. Review formulas, examples, and exports for quicker geometry work.
| Orientation | Center | Vertex | Focus | a | b | c | Standard Form |
|---|---|---|---|---|---|---|---|
| Horizontal | (1, -2) | (6, -2) | (4, -2) | 5 | 4 | 3 | (x - 1)2/25 + (y + 2)2/16 = 1 |
| Vertical | (-3, 1) | (-3, 7) | (-3, 5) | 6 | 4 | 4.472136 | (x + 3)2/16 + (y - 1)2/36 = 1 |
For a horizontal ellipse, a is the distance from the center to a vertex on the x-direction. For a vertical ellipse, a is the distance from the center to a vertex on the y-direction.
c is the distance from the center to a focus. The missing semi-minor axis is found by a2 = b2 + c2, so b = √(a2 − c2).
Horizontal standard form: (x − h)2/a2 + (y − k)2/b2 = 1.
Vertical standard form: (x − h)2/b2 + (y − k)2/a2 = 1.
Eccentricity is e = c/a. Directrices are x = h ± a2/c for horizontal ellipses and y = k ± a2/c for vertical ellipses.
An ellipse is a stretched circle with two focal points. Every point on the curve keeps a constant total distance from both foci. This calculator helps you solve the ellipse when the center, one focus, and one vertex are known. That setup is common in coordinate geometry. It is also useful in graphing, design, and exam practice.
The tool converts geometric inputs into a full analytic description. It finds the semi-major axis, semi-minor axis, and focal distance. It also returns the eccentricity, area, major axis length, and minor axis length. You also get co-vertices, the second focus, the opposite vertex, and directrix equations. The standard form and expanded form are shown clearly. This saves time and reduces algebra mistakes.
The center fixes the ellipse position on the plane. The vertex shows how far the major axis extends from the center. The focus tells how strongly the ellipse is stretched. From these three pieces, the calculator derives the missing semi-minor axis using the relation a² = b² + c². That relationship is the key step in ellipse solving. Once a, b, and c are known, the remaining properties follow directly.
Students can use this page to verify homework steps. Teachers can generate examples for lessons and quizzes. Test takers can quickly review orientation, axis lengths, and equation structure. The export options also help when sharing worked results. The example table makes input patterns easier to understand. The formula section explains each value without heavy wording. This makes the page useful for both learning and checking.
A valid horizontal ellipse needs the focus and vertex on the same horizontal line as the center. A valid vertical ellipse needs the same vertical line. The vertex must be farther from the center than the focus. Otherwise, a becomes smaller than c and the figure is not an ellipse. This calculator checks those conditions before showing results. That makes it helpful for classroom practice, tutoring sessions, and fast self-review. Because outputs are structured neatly, you can compare several cases and spot how axis changes affect eccentricity and directrices.
It solves an ellipse from the center, one focus, and one vertex. It returns the semi-major axis, semi-minor axis, eccentricity, co-vertices, second focus, opposite vertex, directrices, area, and equation forms.
Choose horizontal when the vertex and focus stay on the same y-value as the center. Choose vertical when they stay on the same x-value as the center.
For a real ellipse, c must be smaller than a. When c equals or exceeds a, b becomes zero or imaginary, so the curve is not a valid ellipse.
Yes. The calculator accepts positive, negative, and decimal coordinates. It uses the center as the reference point, so shifted ellipses are handled directly.
The key relation is a² = b² + c². After finding a from the vertex and c from the focus, the calculator computes b = √(a² − c²).
Yes. After calculation, use the CSV button to save the result table. Use the PDF button to open the browser print dialog and save a PDF copy.
A horizontal ellipse centered at (h, k) uses (x − h)²/a² + (y − k)²/b² = 1. The larger denominator stays under the major-axis variable.
Directrices help describe eccentricity and geometric structure. For a horizontal ellipse, they are vertical lines. For a vertical ellipse, they are horizontal lines.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.