Question

If $\frac{5z_1}{7z_2}$ is purely imaginary then the value of $\left|\frac{2z_1+3z_2}{2z_1-3z_2}\right|$ is

Answer 1

Let $\frac{z_1}{z_2}=ib,b\in R$
Now ,$\left|\frac{2z_1+3z_2}{2z_1-3z_2}\right|=\left|\frac{2\frac{z_1}{z_2}+3}{2\frac{z_1}{z_2}-3}\right|$
$=\left|\frac{2ib+3}{2ib-3}\right|$
$=\frac{|2bi+3|}{|2bi-3|}=1$ $[\because \left|\frac{z_1}{z_2}\right|=\frac{|z_1|}{|z_2|}]$