$\begin{array}{1 1}(A)\;\frac{10}{3} \\(B)\;\frac{3}{5} \\(C)\;\frac{6}{5} \\(D)\;\frac{5}{3} \end{array}$

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Let the centre of the circle be (1,h)

Circle touches x axis at (1,0)

Let the circle passes through the point $B(2,3)$

$\therefore CA=CB$

=>$CA^2=CB^2$

=> $(1-1)^2+(h-0)^2$

=> $(1-2)^2+(h-3)^2$

=> $h^2=1+h^2+9-6h$

=> $h= \large\frac{10}{6}$

$h=\large\frac{5}{3}$

This diameter is $2h=\large\frac{10}{3}$

Hence A is the correct answer.

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