This question already has an answer here:
I'm considering an arbitrary, non-degenerate ellipse here, i.e., without assuming that it's centred on the origin or either axis, nor oriented at any specific angle.
I know either 5 points on the perimeter $(x_1,y_1)$ through $(x_5,y_5)$, or the general cartesian equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ where $F = 0$ or $1$. The two forms are equivalent as it's easy to solve the simultaneous equations for 5 values of $(x,y)$ to get $A$ through $F$ when the ellipse isn't degenerate. These uniquely define one ellipse - or a conic at any rate ;-)
But what I'm really after are the cartesian foci and its eccentricity (I can get everything else I might need from these: the major/minor axes, rotate the ellipse to align with the axes, or find its equation in polar form if needed).
I can't find a way to do that......