Ellipse Standard Form
Convert an ellipse equation from general form Ax^2 + By^2 + Cx + Dy + F = 0 to standard form (x-h)^2/a^2 + (y-k)^2/b^2 = 1. Free online conic sections calculator with step-by-step conversion.
General form: Ax² + By² + Cx + Dy + F = 0
About This Calculator
The Ellipse Standard Form Calculator converts equations from general form (Ax^2 + By^2 + Cx + Dy + F = 0) to standard form ((x-h)^2/a^2 + (y-k)^2/b^2 = 1). This conversion uses the completing-the-square method and reveals the ellipse's center (h,k), semi-major axis a, and semi-minor axis b.
Ellipse standard form is essential for graphing ellipses, identifying key properties, and understanding conic sections in algebra, precalculus, and analytic geometry courses.
Frequently Asked Questions
What is the standard form of an ellipse?
The standard form of an ellipse centered at (h,k) is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where a is the semi-major axis and b is the semi-minor axis. For a horizontal ellipse, a > b; for a vertical ellipse, b > a.
How do you convert general form to standard form?
The calculator uses completing the square: group x and y terms, complete the square for each, move constants to the right side, then divide to get 1 on the right. The center is at h = -C/(2A), k = -D/(2B).
Is this calculator free?
Yes, all calculators on Calculy are completely free to use.