We have General Equation of Circle is given as
x2+y2+2gx+2fy+c=0 ---(1)
Since, Circle passes through, (0,0) and (1,0)
Therefore Equation (1) reduces to,
c=0 ---(2)
1+2g+c=0 ---(3),
From (2) and (3), g= -1/2
Also, From As per Question,
Cicle in equation, (1) touches, another circle having equation,
x2+y2=(3)2 ---(4)
SInce the center (0,0) of circle (4) lies on the loci of the circle (1), therefore, The radius of the circle (4) must be equal to diameter of circle(1)
hence, 2√g2+f2-c= 3
=> √1/4+f2=3/2
=> f2= 9/4-1/4
=> f2= 2
=> f= ±√2
Hence Centres of Circle (1) is (-g,-f) or (1/2,±√2)
Option (B) is correct