Given, two equation of circles x2+y2−16x−20y+164=r2 and x2+y2−8x−14y+29=0 Here, C1(8,10),r1=√64+100−164+r2=r and C2(4,7),r2=√16+49−29=6 If two circles with centre C1 and C2 and radius r1 and r2, respectively intersect at two points, then |r1−r2|<C1C2<r1+r2
⇒|r−6|<√(8−4)2+(10−7)2<r+6
|r−6|<5<r+6 When r>6, then r−6<5 and r+6>5 r<11 and r>1 ∴ The maximum integral value of r=10 When r<6, then 6−r<5,r+6>5 6−5<r,r>11 1<r. ∴r is not possible in this case. ∴ The maximum integral value of r is 10 .