The locus of midpoint of the focal chord of curve  $xy=2$  is a conic whose length of latus rectum is  $\lambda$, then  ${\lambda }^2$  equals 

Let midpoint of chord be  $(h,k)$ 
Equation of chord:  $T=S_1$
 $xh+yk=2hk$
This chord passes through the focus of  $xy=2$
Which is  $(2,2)$   or  $(-2,-2)$
$\Rightarrow hk=h+k$   or  $hk=-h-k$
$\Rightarrow xy-x-y=0$  or  $xy+x+y=0$
 $\Rightarrow (x-1)(y-1)=1$  or  $(x+1)(y+1)=1$
This is a rectangular hyperbola whose latus rectum is  $2\sqrt{2}C$  where c=1
 $\Rightarrow \lambda =2\sqrt{2}\Rightarrow {\lambda }^2=8$