The lengh of the transverse common tangent of the circles $x^2+y^2-2x+4y+4=0$ and 

$x^2+y^2+4x-2y+1=0\text{ is }$

$C_1=(1-2),r_1=\sqrt{1+4-4}=1;C_2=(-2,1),r_2=\sqrt{4+1-1}=2.d=C_1{C}_2=\sqrt{9}+9=\sqrt{18}$ 

 The length of transverse  common  tangent $=\sqrt{d^2-{\left(r_1+r_2\right)}^2}$

$=\sqrt{18-(1+2{)}^2}=\sqrt{18-9}=\sqrt{9}=3$