The range of values of $\lambda$for which the circles

$x^2+y^2=4$ and $x^2+y^2-4\lambda x+9=0$ have two common tangents, is

The coordinates of the centers and radii of the two circles are:

Centers: $C_1(0, 1)C_2(2\lambda , 0)$

Radii: $r_1=2r_2=\sqrt{4{\lambda }^2-9}, \lambda \in (-\mathrm{\infty },-3/2)\cup (3/2, \mathrm{\infty })$

The two circles will have exactly two common tangents, if

$\begin{array}{l}\left|r_1-r_2\right|<C_1{C}_2<r_1+r_2\\ \Rightarrow \left|2-\sqrt{4{\lambda }^2-9}\right|<\left|2\lambda \right|<2+\sqrt{4{\lambda }^2-9}\\ \Rightarrow \left.\left|2\lambda \right|<2+\sqrt{4{\lambda }^2-9}\left[\because \left|2-\sqrt{4{\lambda }^2-9}\right|<\left|2\lambda \right|\text{ for all }\lambda \in R\right]\right]\\ \Rightarrow \left|2\lambda \right|-2<\sqrt{4{\lambda }^2-9}\\ \Rightarrow \left(\right|2\lambda |-2{)}^2<4{\lambda }^2-9[\because \left|2\lambda \right|-2>0]\\ \Rightarrow -8\left|\lambda \right|<-13\\ \Rightarrow \left|\lambda \right|>\frac{13}{8}\Rightarrow \lambda \in (-\mathrm{\infty },-13/8)\cup (13/8,\mathrm{\infty })\end{array}$