JEE Advanced 2019 — Paper 1 — Question 44

Let the point $B$ be the reflection of the point $A(2,3)$ with respect to line $8 x-6 y-23=0$. 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 $\Gamma_{\mathrm{B}}$ 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 $A C$ is $\_\_\_\_$ -