$|z_{1}|=5, |z_{2}|=\sqrt{5}$, the argument difference $\alpha$ of complex numbers $z_{1}, z_{2}$ satisfies $\tan \alpha=\frac{1}{2} \Rightarrow \cos \alpha=\frac{2}{\sqrt{5}}$, so $|z_{1}-z_{2}|^{2}=25+5-2 \cdot 5 \cdot \sqrt{5} \cdot \frac{2}{\sqrt{5}}=10 \Rightarrow |z_{1}-z_{2}|=\sqrt{10}$.
8. Let $z_{1}, z_{2}$ be complex numbers, and satisfy $\left|z_{1}\right|=5, \frac{z_{1}}{z_{2}}=2+\mathrm{i}$ (where $\mathrm{i}$ is the imaginary unit), then the value of $\left|z_{1}-z_{2}\right|$ is