5. Given complex numbers $z_{1}, z_{2}$ satisfy $\left|z_{1}\right|=\left|z_{1}-2 z_{2}\right|, z_{1} \overline{z_{2}}=\sqrt{3}(1-\mathrm{i})$, where $\mathrm{i}$ is the imaginary unit, then $\frac{z_{1}}{z_{2}}=$
$$
\begin{array}{l}
z_{1} \overline{z_{1}}=\left(z_{1}-2 z_{2}\right)\left(\overline{z_{1}}-2 \overline{z_{2}}\right)=z_{1} \overline{z_{1}}-2 z_{1} \overline{z_{2}}-2 z_{2} \overline{z_{1}}+4 z_{2} \overline{z_{2}} \\
\Rightarrow 2 z_{2} \overline{z_{2}}=z_{1} \overline{z_{2}}+z_{2} \overline{z_{1}}=\sqrt{3}(1-\mathrm{i})+\sqrt{3}(1+\mathrm{i})=2 \sqrt{3} \Rightarrow z_{2} \overline{z_{2}}=\sqrt{3},
\end{array}
$$

So $\frac{z_{1}}{z_{2}}=\frac{z_{1} \overline{z_{2}}}{z_{2} \overline{z_{2}}}=1-\mathrm{i}$.