Question

If arg $\left[\frac{z_1}{z_2}\right]$ $=$ $\frac{\pi }{2},$ then find the value of $\left|\frac{z_1+z_2}{z_1-z_2}\right|.$

Answer 1

$arg\left(\frac{z_1}{z_2}\right)=\frac{\pi }{2}$

$\Rightarrow \frac{z_1}{z_2}$ is purely imaginary

Let $\frac{z_1}{z_2}=ki$

Now $\left|\frac{z_1+z_2}{z_1-z_2}\right|=\left|\frac{\frac{z_1}{z_2}+1}{\frac{z_1}{z_2}-1}\right|=\left|\frac{ki+1}{ki-1}\right|$

$\frac{\sqrt{1+k^2}}{\sqrt{1+k^2}}=1$.