The equations of the circles are x2+y2−10x−10y+λ=0 and x2+y2−4x−4y+6=0...(2) C1= centre of (1)=(5,5) C2= centre of (2)=(2,2) d= distance between centres =C1C2=√9+9=√18 r1=√50−λ,r2=√2 For exactly two common tangents we have r1−r2<C1C2<r1+r2 ⇒√50−λ−√2<3√2<√50−λ+√2 ⇒√50−λ−√2<3√2 or 3√2<√50−λ+√2 ⇒√50−λ<4√2 or 2√2<√50−λ ⇒50−λ<32 or 8<50−λ ⇒λ>18 or λ<42 Required interval is (18,42)