The correct option is D g=34 or f=2
The radical axis of the given circles x2+y2+2gx+2fy+c=0 and x2+y2+(32)x+4y+c=0, is (2g−32)x+(2f−4)y=0 or (4g−3)x+4(f−2)y=0 (1)
This radical axis (1) touches the circles x2+y2+2x+2y+1=0, ...(2)
If the length of ⊥ from centre (−1,−1) on the line (1)= radius of circle (2) i.e.,
(4g−3)(−1)+4(f−2)(−1)√[(4g−3)2+16(f−2)2]=±√(1+1−1)
⇒[(4g−3)+4(f−2)]2=(4g−3)2+16(f−2)2
⇒8(4g−3)(f−2)=0
⇒g=34 or f=2