How this calculator works
Opposite vertices, opposite co-vertices and the two foci are symmetric about the centre, so their midpoint gives the centre directly. For a general equation without an xy term, completing each square gives the centre and the two semi-axes. Same-sign squared coefficients alone are insufficient: the completed right side must also give positive axis squares. Standard mode assumes the displayed denominators are positive and reads h and k directly.
Formula
Centre = ((x₁+x₂)/2, (y₁+y₂)/2). For Ax²+By²+Dx+Ey+F=0: h=-D/(2A), k=-E/(2B), R=Ah²+Bk²-F; rx²=R/A, ry²=R/B.
Worked example
For 4x²+9y²-16x+54y+61=0, completing squares gives 4(x-2)²+9(y+3)²=36. The centre is (2,-3), with horizontal semi-axis 3 and vertical semi-axis 2.
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Frequently asked questions
Can I enter two adjacent vertices?
No. Use the two opposite endpoints of one axis, or the two foci. Adjacent axis endpoints do not share the centre as their midpoint.
What about a rotated equation with an xy term?
The equation mode is limited to axis-aligned ellipses. Use a symmetric point pair for a rotated ellipse.
Do two foci determine the whole ellipse?
They determine its centre, but a size measurement is still needed to find its axes. Coincident foci describe the circular case.