Rectangular Hyperbola

A square $ABCD$ has all its vertices on the curve $x^2{y}^2=1.$ The midpoints of its sides also lie on the same curve. Then, the square of area of $ABCD$ is

Three points $A, B$ and $C$ are taken on a rectangular hyperbola $xy=4$, where $B$ and $C$ have co-ordinates $(-2, -2)$ and $\left(6, \frac{2}{3}\right)$ respectively and $A$ lies in the first quadrant. If the normal at $A$ is parallel to $BC$, then

The locus of the foot of the perpendicular from the center of the hyperbola $xy=c^2$ on a variable tangent is