The equation of one of the common tangents of the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{y}+4=0$ and the parabola ${\mathrm{y}}^2=\mathrm{x}$ is

$$\begin{gathered} {\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{y}+4=0 \\ \mathrm{centre}=\left(0, 3\right) \\ \mathrm{r}=\sqrt{0+9-4}=\sqrt{5} \\ {\mathrm{y}}^2=\mathrm{x} \\ \Rightarrow \mathrm{a}=\frac{1}{4} \end{gathered}$$

Equation tangent is $\mathrm{y}=\mathrm{mx}+\frac{1}{4\mathrm{m}}$

$4{\mathrm{m}}^2\mathrm{x}-4\mathrm{my}+1=0$ it also tangent to circle then r = d

$$\begin{gathered} \sqrt{5}=\frac{\left|0-12\mathrm{m}+1\right|}{\sqrt{16{\mathrm{m}}^4+16{\mathrm{m}}^2}} \\ \Rightarrow \mathrm{m}=1/2 \\ Equation of common \tan gent is \mathrm{y}=\frac{\mathrm{x}}{2}+\frac{1}{2} \\ \Rightarrow 2\mathrm{y}=\mathrm{x}+1 \\ \Rightarrow \mathrm{x}-2\mathrm{y}+1=0 \end{gathered}$$