If $\mathrm{y}={\mathrm{m}}_1\mathrm{x}+{\mathrm{c}}_1$ and $\mathrm{y}={\mathrm{m}}_2\mathrm{x}+{\mathrm{c}}_2,{ \mathrm{m}}_1\ne {\mathrm{m}}_2$ are two common tangents of circle $x^2+y^2=2$ and parabola ${\mathrm{y}}^2=\mathrm{x}$, then the value of $8\left|m_1{ \mathrm{m}}_2\right|$ is equal to

$$\begin{gathered} C_1:x^2+y^2=2 \\ {C}_2:y^2=x \end{gathered}$$

Let tangent to parabola be $y=mx+\frac{1}{4m}$.
It is also a tangent of circle so distance from centre of circle (0,0) will be $\sqrt{2}$.

$\left|\frac{\frac{1}{4m}}{\sqrt{1+m^2}}\right|=\sqrt{2}\Rightarrow 1=32m^2+32m^4$

by solving

$$\begin{gathered} m^2=\frac{3\sqrt{2}-4}{8}, m^2=\frac{-3\sqrt{2}-4}{8}\text{ (rejected) } \\ m=\pm \sqrt{\frac{3\sqrt{2}-4}{8}} \\ \text{ so, }8\left|{ \mathrm{m}}_1{ \mathrm{m}}_2\right|=3\sqrt{2}-4 \end{gathered}$$