Consider a family of circles which are passing through the point $(– 1, 1)$ and are tangent to $x-$axis. If $(h, k)$ are the coordinates of the centre of the circles, then the set of values of $k$ is given by the interval

Equation of the circle with centre $(h, k)$ and touching $x-$axis is
                  $(x-h{)}^2+(y-k{)}^2=k^2$   
which passes through $(– 1, 1)$    
if              $h^2+2h-2k+2=0$
or if        $(h+1{)}^2=2k-1$
For real values of $h,2k-1\ge 0$.
$\Rightarrow k\ge 1/2$