How this calculator works
Standard form locates the centre and the two semi-axes. In coefficient mode, completing the square groups the x and y terms; the table shows the resulting denominators. Vertex and focus modes use the midpoint of the selected point pair as the centre. This calculator handles horizontal or vertical axes. It excludes equations with an xy term, which require rotation.
Formula
(x-h)²/rx² + (y-k)²/ry² = 1; a = max(rx,ry); b = min(rx,ry); c² = a²-b²; e = c/a. For Ax² + Cy² + Dx + Ey + F = 0: h = -D/(2A), k = -E/(2C), Q = Ah²+Ck²-F, rx² = Q/A, ry² = Q/C.
Worked example
9x² + 25y² - 18x - 100y - 116 = 0 becomes 9(x-1)² + 25(y-2)² = 225. Its standard form is (x-1)²/25 + (y-2)²/9 = 1, with foci (-3,2) and (5,2).
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Frequently asked questions
Which axis gets a²?
The major axis. A vertical major axis puts a² below the y term, while the x term gets b².
Can both squared coefficients be negative?
Yes. Multiplying an entire equation by -1 leaves its ellipse unchanged. The completed squared-axis values must still be positive.
Can I enter rotated vertices?
No. Use point pairs on a horizontal or vertical line. A rotated ellipse requires an xy term and a different transformation.
What happens for a circle?
Equal semi-axes are allowed and both foci coincide at the centre. To specify that case directly, use centre and semi-axes mode.