Mathematics

Problem

Let the point

B

be the reflection of the point

A(2, 3)

with respect to the line

8x - 6y - 23 = 0

. Let

\Gamma_A

and

\Gamma_B

be circles of radii 2 and 1 with centres

A

and

B

respectively. Let

T

be a common tangent to the circles

\Gamma_A

and

\Gamma_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 ___.