There are several approaches. One approach is the following:

Since area is invariant under rotation and translation, an ellipse of the form can be assumed WLOG. The problem is to find the area of (I assume) the largest square that can fit in this ellipse.

To get the coordinates of the corners of the square (and hence the length of a side and hence the area), solve

.... (1)

.... (2)

simultaneously:

.

Hence the largest square has sidelength .

Hence the area of the largest square is .