= Quadratic-form ellipse area
{title2=$\mathcal A=\pi R/\sqrt{\det M}$}
For a two-dimensional <symmetric positive-definite matrix> $M$ and $R>0$, the region $\mathbf x^TM\mathbf x\leq R$ is a filled <ellipse> of <area> $\pi R/\sqrt{\det M}$. By the <spectral theorem for real symmetric matrices>, an <orthogonal change of coordinates> gives positive <eigenvalues> $\lambda_1,\lambda_2$ and semiaxes $\sqrt{R/\lambda_1}$ and $\sqrt{R/\lambda_2}$. Their product and the <area of an ellipse> prove the formula. Translating the <center of an ellipse> leaves the <area> unchanged.
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