Let ` z_1 and z_2` be complex numbers of such that `z_1!=z_2 and |z_1|=|z_2|. If z_1` has positive real part and `z_2` has negative imginary part, then which of the following statemernts are correct for te vaue of `(z_1+z_2)/(z_1-z_2)`

Options

A. zero
B. real and positive
C. real and negative
D. purely imaginary

Answer

A, D

Explanation

Given `|z_1|=|z_2|`
Now `(z_1+z_2)/(z_1-z_2)xx(barz_1-barz_2)/(z_1-z_2)=(z_1barz_1-z_1barz_2+z_2barz_1-z_2barz_2)/(|z_1-z_2|)`
`=(|z|^2+(z_2barz_1-z_1barz_2)-|z_2|^2)/(|z_1-z_2|^2)`
`=(z_2barz_1-z_1barz_2)/(|z_1-z_2|^2)" "[because |z_1|^2=|z_2|^2]`
As we know `z-bar z `=2 i Im (z)
`therefore ( z_2barz_1-z_1barz_2=2i Im (z_2 bar z_1)`
`therefore (z_1+z_2)/(z_1-z_2)=(2i Im(z_2 bar z_1))/(|z_1-z_2|^2)`
Which is purely imaginary or zero.