Question

If ${2z}_1{/3z}_2$ is a purely imaginary number, then find the value of $\left|(z_1-z_2)/(z_1+z_2)\right|$.

Answer 1

Since, 2z1/3z2 = k'i ...(Given)
or, z1/z2= (3k'/2) i
or, z1/z2= ki (where 3k'/2=k) ...(1)
|(z1-z2)/ (z1+z2)| = |z1{1-(z2/z1)}/z1{1+(z2/z
Cancelling z1's, above equation becomes
|(z1-z2)/ (z1+z2)| = |[1-(z2/z1)]/[1+(z2/z1)]|
= | (1-ki)/ (1+ki) | ...(using (1))
= |(1-ki)|/ |(1+ki)|
= 1