The equation of the circles are

${x^2} + {y^2} – 10x – 10y + \lambda  = 0\,\,\,\,\,\,\,……\left( 1 \right)$

and ${x^2} + {y^2} – 4x – 4y + 6 = 0\,\,\,\,\,\,\,……\left( 2 \right)$

${C_1} = \,$ center of $\left( 1 \right) = \left( {5,5} \right)$

${C_2} = \,$ center of $\,\left( 2 \right) = \left( {2,2} \right)$

$ = {C_1}{C_2} = \sqrt {9 + 9}  = \sqrt {18} $

${r_1} = \sqrt {50 – \lambda } ,{r_2} = \sqrt 2 $

For exactly two common tangents we have

${r_1} – {r_2} < {C_1}{C_2} < {r_1} + {r_2}$

$ \Rightarrow \sqrt {50 – \lambda }  – \sqrt 2  < 3\sqrt 2  < \sqrt {50 – \lambda }  + \sqrt 2 $

$ \Rightarrow \sqrt {50 – \lambda }  – \sqrt 2  < 3\sqrt 2 $ or $\,3\sqrt 2  < \sqrt {50 – \lambda }  + \sqrt 2 $

$ \Rightarrow \sqrt {50 – \lambda }  < 4\sqrt 2 $ or $2\sqrt 2  < \sqrt {50 – \lambda } $

$50 – \lambda  < 32$ or $8 > 50 – \lambda $

$ \Rightarrow \lambda  < 18$ or $\lambda  < 42$

Required interval is $(18,42)$