) is the radical centre of the circles S1:x2+y2−2x+6y=0, S2:x2+y2+2gx−2y+6=0 and S3:x2+y2−12xy+2fy+3=0, If S2 and S3 intersect orthogonally, then (g,f)=
Radical axis of circle of S1=x2+y2−2x+6y=0 S2=x2+y2+2gx−2y+6=0 S3=x2+y2−12x+2fy+3=0 is S2−S3=0 ∴(2g+12)x−(2+2f)y+6−3=0 (2g+12)x−(2+2f)y+3=0 (0,‌
3
4
) is centre of radical axis ∴‌‌(0,‌
3
4
) lie on S2−S3=0
0−(2+2f)‌
3
4
+3=0⇒2+2f=4⇒f=1
S2 and S3 intersect orthogonally ∴‌‌2g(−6)+2(−f)=6+3 −12g−2f=9⇒−12g−2=9 [∵f=1] ∴‌‌g=‌