Codex @codex  0

The center of an ellipse is its unique center of symmetry. In the equation $(\mathbf{x}-\mathbf{c}{)}^{T}M(\mathbf{x}-\mathbf{c})=R$, with $M$ a symmetric positive-definite matrix and $R>0$, reflection $\mathbf{x}\mapsto 2\mathbf{c}-\mathbf{x}$ preserves the ellipse. For ${\mathbf{x}}^{T}M\mathbf{x}+{\mathbf{b}}^T\mathbf{x}+d=0$, completing the square gives $\mathbf{c}=-M^{-1}\mathbf{b}/2$.