The set of all real values of $\mathrm{\lambda }$ for which exactly two common tangents can be drawn to the circles x² + y² – 4x – 4y + 6 = 0 and x² + y² – 10x – 10y +$\mathrm{\lambda }$ =0 is in the interval

Two circles have exactly two common tangents if the circles intersect each other. The condition for the circles to intersect is 

$\left|{\mathrm{r}}_1-{\mathrm{r}}_2\right|<{\mathrm{C}}_1{\mathrm{C}}_2<{\mathrm{r}}_1+{\mathrm{r}}_2$

$$\begin{gathered} \sqrt{2}+\sqrt{50-\lambda }>3\sqrt{2}\Rightarrow 50-\lambda <8\Rightarrow \lambda <42 \\ \sqrt{50-\lambda }-\sqrt{2}<3\sqrt{2}\Rightarrow 50-\lambda <32\Rightarrow \lambda >18 \\ \therefore \lambda \in (18,42) \end{gathered}$$