If these two points lie anywhere on the circumference but diametrically opposite each other, then the radius of the circle won't be minimum. Thus, the two points (1,0) and (0,1) form the end-points of diameter of the circle with minimum radius.
Equation of circle with diameter end-points (x1,y1)(x1,y1) and (x2,y2)(x2,y2) is:
Let equation of circle is
This passes through (1,0) and (0,1) means 1+2g+c=0 and 1+2f+c=0
=>g=f and c=-2g-1
Radius R of circle is given by
Thus S is minimum for g=-1/2 .
Hence equation of required circle is