Since chord $\mathit{C}\mathit{D}$ and tangent at point $\mathit{T}$ intersect each other at $\mathit{P}$ ,
$\therefore \mathrm{PC}\times \mathrm{PD}={\mathrm{PT}}^2$ .........(1)
Since chord $\mathit{A}\mathit{B}$ and tangent at point $\mathit{T}$ intersect each other at $\mathrm{P}$ ,
$\therefore \mathrm{PA}\times \mathrm{PB}={\mathrm{PT}}^2$ ..........(2)
From (1) and (2),
$\mathrm{PC}\times \mathrm{PD}=\mathrm{PA}\times \mathrm{PB}$ .........(3)
Given : $\mathrm{CD}=7.8\mathrm{cm};\mathrm{PD}=5\mathrm{cm},\mathrm{PB}=$ $4\mathrm{cm}$.
$\therefore \mathrm{PA}=\mathrm{PB}+\mathrm{AB}=4+\mathrm{AB},$
$\mathrm{PC}=\mathrm{PD}+\mathrm{CD}=5+7.8=12.8\mathrm{cm}.$
Putting these values in eq. (3)
$12.8\times 5=\left(4+\mathrm{AB}\right)\times 4$
$\Rightarrow 4+\mathrm{AB}=\frac{12.8\times 5}{4}$
$\Rightarrow 4+\mathrm{AB}=16$
$\Rightarrow \mathrm{AB}=12\mathrm{cm}.$