If `z_(1)` and `z_(2)` are two complex numbers satisying the equation.
`|(iz_(1)+z_(2))/(iz_(1)-z_(2))|=1`, then `z_(1)/z_(2)` is

Options

A. 0
B. purely real
C. negative real
D. purely imaginary

Answer

D

Explanation

We have,
`(|iz_(1)+z_(2)|)/(|iz_(1)-z_(2)|)=1`
`rArr |iz_(1)+z_(2)|^(2)=|iz_(1)-z_(2)|^(2)`
`rArr (iz_(1)+z_(2))(-ibarz_(1)+barz_(2))=(ibarz_(1)-z_(2))(-ibarz_(1)-barz_(2))`
`rArr z_(1)barz_(1)+iz_(1)barz_(2)-iz_(2)barz_(1)+z_(2)barz_(2)=z_(1)barz_(1)-iz_(1)barz_(2)+ibarz_(1)z_(2)+z_(2)barz_(2)`
`rArr z_(1)barz_(2)=(bar(z_(1)barz_(2)))`
`rArr z_(1)barz_(2)` is purely imaginary.
`rArr z_(1)/z_(2)|z_(2)|^(2)` is purely imaginary.
`rArr z_(1)/z_(2)` is purely imaginary.