Question

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

Answer 1

We have $z_1=2-i,z_2=1+i$
$\therefore \left|\frac{z_1+z_2+1}{z_1-z_2+1}\right|=\left|\frac{(2-i)+(1+i)+1}{(2-i)-(1+i)+1}\right|$
$=\left|\frac{4}{2-2i}\right|=\left|\frac{4}{2(1-i)}\right|$
$=|\frac{2}{1-i}\times \frac{1+i}{1+i}|=\left|\frac{2(1+i)}{1^2-i^2}\right|$
$=\left|\frac{2(1+i)}{1+1}\right|=\left|\frac{2(1+i)}{2}\right|$
$=|1+i|=\sqrt{1^2+1^2}=\sqrt{2}$