Consider a family of circles which are passing through  M1,1 and are tangent to  x-axis.  If  h,k is the centre of circle, then:

Equation of family of circles with centre (h,k) and touching x -axis is: x−h2+y−k2=k2⇒x2+y2−2hx−2ky+h2=0 As M(1,1) lies on it ⇒  2−2h−2k+h2=0⇒   k=h−12+12≥12∴k≥12