The number of tangents that can be drawn from the point $\left(\frac{5}{2},1\right)$to the circle passing through the points $(1,\sqrt{3})(1,-\sqrt{3})$ and $(3,-\sqrt{3})$ is

Given points $\mathrm{A}=(1, \sqrt{3}), \mathrm{B}=(1, -\sqrt{3}), \mathrm{C}=(3, -\sqrt{3})$ since AB is parallel to y-axis and BC is parallel to x-axis then $\angle \mathrm{ABC}={90}^0$

$\Rightarrow \mathrm{AC}$ is diameter of circle 

Now equation of circle is 

$$\begin{gathered} (\mathrm{x}-1) (\mathrm{x}-3)+(\mathrm{y}-\sqrt{3}) (\mathrm{y}+\sqrt{3})=0 \\ \mathrm{S}\equiv {\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}=0- \left(1\right) \end{gathered}$$

Given point $\mathrm{P}=\left(\frac{5}{2}, 1\right)$

Now ${\mathrm{S}}_{11}=\frac{25}{4}+1-4\left(\frac{5}{2}\right)$

                 $=\frac{-11}{4}<0$

$\therefore \left(\frac{5}{2}, 1\right)$ lies inside the circle 

$\therefore$ No tangent can be drawn from the point $\left(\frac{5}{2}, 1\right)$ to the circle (1)