Let the point B be the reflection of the point A(2,3) with respect to the line $8x-6y-23=0.$ Let $Γ_{A}$ and $Γ_{B}$ be circles of radii 2 and 1 with centers A and B respectively. Let T be a common tangent to the circles $Γ_{A}$ and $Γ_{B}$ such that both the circles are on the same side of T. If C is the point of intersection of T and the line passing through A and B, then the length of the line segment AC is ____.

$\text{Given: } A(2,3),\quad B = \text{reflection of } A \text{ in } 8x-6y-23=0$

$\Gamma_A: \text{centre } A,\ r = 2;\qquad \Gamma_B: \text{centre } B,\ r = 1$

$T: \text{common tangent, both circles on the same side}$

$C = T \cap AB;\qquad \text{find } AC$

$L: 8x - 6y - 23 = 0$

$A(2,3)$

$d(A,L) = \frac{|8(2)-6(3)-23|}{\sqrt{8^2+6^2}} = \frac{25}{10} = \frac{5}{2}$

$M = \left(4, \frac{3}{2}\right)\ (\text{foot of perpendicular})$

$B = 2M - A = 2\left(4, \frac{3}{2}\right) - (2,3) = (6,0)$

$AB = 2\,d(A,L) = 2\cdot\frac{5}{2} = 5$

$\Gamma_A: (x-2)^2 + (y-3)^2 = 4$

$\Gamma_B: (x-6)^2 + y^2 = 1$

$1.\ \text{reflect } A \text{ in } L \Rightarrow B$

$2.\ \text{direct common tangent meets } AB \text{ at } C$

$3.\ \frac{CA}{CB} = \frac{r_A}{r_B} \Rightarrow AC$

$\triangle C A T_A \sim \triangle C B T_B\ (\text{common } \angle C,\ \text{right angles at tangency})$

$\Rightarrow\ \frac{CA}{CB} = \frac{r_A}{r_B} = \frac{2}{1}$

$C = T \cap AB,\quad C \text{ lies beyond } B$

$\frac{AC}{CB} = \frac{r_A}{r_B} = \frac{2}{1}$

$AC - CB = AB = 5$

$AC = 2\,CB\ \Rightarrow\ 2CB - CB = 5\ \Rightarrow\ CB = 5$

$\Rightarrow\ AC = 2\cdot 5 = 10$

$C = 2B - A = 2(6,0)-(2,3) = (10,-3)$

$AC = \sqrt{(10-2)^2 + (-3-3)^2} = \sqrt{64+36} = 10$

$\boxed{AC = 10}$