Equation of a common tangent to the circle,

Let equation of tangent to the parabola $y^{2}=4 x$ is

$y=m x+\frac{1}{m}$

$\Rightarrow m^{2} x-y m+1=0$ is tangent to $x^{2}+y^{2}-6 x=0$

$\Rightarrow \frac{\left|3 m^{2}+1\right|}{\sqrt{m^{4}+m^{2}}}=3$

$m=\pm \frac{1}{\sqrt{3}}$

$\Rightarrow$ tangent are $x+\sqrt{3} y+3=0$

and $x-\sqrt{3} y+3=0$