## Solution :

Let us assume that the coordinates of the center of the circle are C(h,k) and its radius is r.

Now, since the circle touches X-axis at (1,0), hence its radius should be equal to ordinate of center.

\(\implies\) r = k

Hence, the equation of circle is \((x – h)^2 + (y – k)^2\) = \(k^2\)

Also, given that the circle passes through points (1, 0) and (2, 3). Hence, substituting them, in the equation of circle we get

\((1 – h)^2 + (0 – k)^2\) = \(k^2\) ……(i)

\((2 – h)^2 + (3 – k)^2\) = \(k^2\) ……(ii)

from equations (i) and (ii), we get

k = \(5\over 3\)

Hence, The diameter of the circle is 2k = \(10\over 3\)

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