Question

If $z_1=2-i,z_2=1+i,\mathrm{find}\left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|$

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ {z}_1=2-i,z_2=1+i \\ \therefore \left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|=\left|\frac{2-i+1+i+1}{2-i-1-i+i}\right| \\ =\left|\frac{4}{1-i}\right| \\ =\frac{4}{\left|1-i\right|} \\ \mathrm{Also},\left|1-i\right|=\sqrt{1^2+i^2}\left(\because \left|a+bi\right|=\sqrt{a^2+b^2}\right) \\ =\sqrt{2} \\ \therefore \left|\frac{z_1+z_2+1}{z_1-z_2+i}\right|=\frac{4}{\sqrt{2}} \end{gathered}$$