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Is it possible to get the area of an ellipse from the foci alone? Or do I need at least one point on the ellipse too?

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up vote 4 down vote accepted

If the sum of the distance to the foci is just the distance between the foci, the ellipse shrinks to a line. If it is bigger, then the area is strictly positive.

This tells you that the foci alone are not sufficient to know the area. A point on the ellipse will be sufficient, but of course other information will also be sufficient.

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If the foci are points $p,q\in\mathbb{R}^{2}$ on a horizontal line and a point on the ellipse is $c\in\mathbb{R}^{2}$, then the string length $\ell=\left|p-c\right|+\left|q-c\right|$ (the distance from the first focus to the point on the ellipse to the second focus) determines the semi-axis lengths. Using the Pythagorean theorem, the vertical semi-axis has length $\sqrt{\frac{\ell^{2}}{4}-\frac{\left|p-q\right|^{2}}{4}}$. Using the fact that the horizontal semi-axis is along the line joining $p$ to $q$, the horizontal semi-axis has length $\frac{\ell}{2}$. Thus the area is $\pi\sqrt{\frac{\ell^{2}-\left|p-q\right|^{2}}{4}}\frac{\ell}{2}$ ($\pi$ times each semi-major axis length, analogous to the circle area formula $\pi r^{2}$).

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