Let `z_1a n dz_2` be complex numbers such that `z_1!=z_2` and `|z_1|=|z_2|dot` If `z_1` has positive real part and `z_2` has negative imaginary part, then `(z_1+z_2)/(z_1-z_2)` may be (a)zero (b) real and positive (c)real and negative (d) purely imaginary

Options

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

Answer

A, D

Explanation

Let `z_(1) =a + ib, a gt 0` and `b in R: z_(2)` = c =id, `d gt 0, c in R`
Given `|z_(1)| = |z_(2)|`
`rArr a^(2) + b^(2) = c^(2) + d^(2)`
`rArr a^(2) - c^(2) =d^(2) -b^(2)`
Now`,(z_(1) +z_(2))/(z_(1)-z_(2))=((a+c)+i(b+d))/((a-c)+i(b-d))`
`[(a^(2) -c^(2) )+(b^(2) -d^(2))]+=(i[(a-c)(b+d)-(a+c)(b-d))]/((a-c)^(2) + (b-d)^(2))`
This value will be purely imaginary as `a^(2)-c^(2) =d^(2) - b^(2)` or the value of will be zero in case `a+ c = b + d =0`