If (4,-6) is a centre of similitude for the circles $x^2+y^2=4$  and $x^2+y^2+4x-6y+4=0$ . Then the length of common tangent of the circle is

$C_1=\left(0,0\right),r_1=2$

$C_2=\left(-2,3\right),r_2=3$

$d=C_1{C}_2=\sqrt{13}$

  Clearly $\left(4,-6\right)$  is an external centre of similitude

$\therefore$  length of direct common tangent $=\sqrt{d^2-{\left(r_1-r_2\right)}^2}=\sqrt{12}$