# The length of the diameter of the circle which touches the x-axis at the point $(1,0)$ and passes through the point $(2,3)$ is

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

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.