Minimum area of the circle which touches the parabolas  $y=x^2+1$  and  $y^2=x-1$ is

$y=x^2+1\&y^2=x-1$  are symmetrical about  $y=x$
$\therefore$  tangent at A is parallel to  $y=x$
 $\frac{dy}{dx}=2x\Rightarrow 2x=1\Rightarrow x=\frac{1}{2},y=\frac{5}{4}$
 $A=(\frac{1}{2},\frac{5}{4}),B=(\frac{5}{4},\frac{1}{2})$
 $\frac{1}{2}\left(AB\right)=\frac{3}{4\sqrt{2}}=radius$