Question:
If $z_{1}=2-i, z_{2}=1+i$, find $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|$
Solution:
Given:
$z_{1}=2-i, z_{2}=1+i$
$\therefore\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|=\left|\frac{2-i+1+i+1}{2-i-1-i+i}\right|$
$=\left|\frac{4}{1-i}\right|$
$=\frac{4}{|1-i|}$
Also, $|1-i|=\sqrt{1^{2}+i^{2}} \quad\left(\because|a+b i|=\sqrt{a^{2}+b^{2}}\right)$
$=\sqrt{2}$
$\therefore\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+i}\right|=\frac{4}{\sqrt{2}}$