Center of Ellipse Calculator — General Conic Form
Enter the six coefficients A–F of the general conic equation and find the center (h, k) of the ellipse, with a step-by-step derivation using partial derivatives.
h = 3, k = 2
- 1
Determinant Δ = 4AC − B²
4 × 1 × 2 − 0² = 8 - 2
Numerator for h: −2C·D + B·E
−2 × 2 × -6 + 0 × -8 = 24 - 3
Center x-coordinate h = numerator ÷ Δ
24 ÷ 8 = 3This is the x-coordinate of the ellipse center.
How does this calculator work?
For conic Ax² + Bxy + Cy² + Dx + Ey + F = 0 with B² − 4AC < 0 (ellipse), the center is h = (−2CD + BE)/(4AC − B²), k = (−2AE + BD)/(4AC − B²). For axis-aligned ellipses (B = 0) this simplifies to h = −D/(2A), k = −E/(2C). Example: x² + 2y² − 6x − 8y + 15 = 0 → center (3, 2).
Formula
How this is calculated
Any ellipse can be written in the general conic form Ax² + Bxy + Cy² + Dx + Ey + F = 0 where B² − 4AC < 0. The center of the ellipse is the point (h, k) where the conic is symmetric — geometrically, all chords through it are bisected there. Algebraically, it is where both first-order partial derivatives vanish simultaneously: ∂f/∂x = 2Ah + Bk + D = 0 and ∂f/∂y = Bh + 2Ck + E = 0.
This gives a 2×2 linear system in h and k. Solving by Cramer's rule uses the determinant Δ = 4AC − B², which is positive when B² − 4AC < 0 (i.e. for a real ellipse), guaranteeing a unique solution. For axis-aligned ellipses (B = 0), the system decouples to h = −D/(2A) and k = −E/(2C) — a useful special case. When B ≠ 0 the ellipse is tilted and the full formula is needed.
Once the center is found, you can shift coordinates to (x′, y′) = (x − h, y − k), which removes the linear terms and gives the ellipse in translated standard form. The semi-axes are shown automatically for axis-aligned cases (B = 0); for tilted ellipses you would also need to rotate to remove the Bxy cross term.
Frequently asked questions
If the equation is already in the form (x − h)²/a² + (y − k)²/b² = 1, the center is simply (h, k). Expand this into the general form to use this calculator, or just read h and k directly from the standard form.
The sign of the discriminant B² − 4AC determines the conic type. Negative means an ellipse (or circle if also A = C, B = 0), zero means a parabola, and positive means a hyperbola. This calculator only handles ellipses (negative discriminant).
First find the center (h, k) with this calculator. Then translate the origin to (h, k) by substituting x = x′ + h, y = y′ + k, which removes the D and E terms. If B ≠ 0, rotate axes by angle θ = ½ arctan(B/(A − C)) to eliminate the cross term, then read off a and b from the resulting equation.
Also known as
TG we-Calculate Editorial Team. (2026). Center of Ellipse Calculator — General Conic Form [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/center-of-ellipse-calculator
TG we-Calculate Editorial Team. "Center of Ellipse Calculator — General Conic Form." TG we-Calculate. 2026. https://we-calculate.com/calculator/center-of-ellipse-calculator.
TG we-Calculate Editorial Team, "Center of Ellipse Calculator — General Conic Form," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/center-of-ellipse-calculator
@misc{wecalculate_center_of_ellipse_calculator, title = {Center of Ellipse Calculator — General Conic Form}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/center-of-ellipse-calculator}}, year = {2026}, note = {TG we-Calculate} }
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