The inverse point of $(1, -1)$with respect to $x^2+y^2=4$, is

We know that two points are inverse points with respect to a circle if each lies on the polar of the other. The polar of (1, -1) with respect to $x^2+y^2=4$ is  $x-y=4$

Clearly, (2, - 2) lies on it. Hence, the inverse point of (1, -1) with respect of $x^2+y^2=4$ is  (1,-1).