Let equation of circle touches the positive coordinate axes having radius r is (x−r)2+(y−r)2=r2...(i) ∵ The circle (i) touches the another given circle x2+y2−12x−10y+52=0 externally then c1c2=r1+r2 Where c1(r,r),c2(6,5),r1=r and r2=3 ∴√(r−6)2+(r−5)2=r+3 ⇒2r2−22r+61=r2+6r+9
⇒r2−28r+52=0⇒(r−26)(r−2)=0
⇒r=2 or 26 ∴ There are two possible circles having centres (2,2) and (26,26) respectively. So, required distance between centres