If `z_(1),z_(2)` are tow complex numberes `(z_(1) ne z_(2))` satisfying `|z_(1)^(2)- z_(2)^(2)|=|barz_(1)^(2)+barz_(2)^(2) - 2barz_(1)barz_(2)|`, then

Options

A. `(z_(1))/(z_(2))` is purely imaginary
B. `(z_(1))/(z_(2))` is purely real
C. `|argz_(1) - argz_(2)| = pi`
D. `|arg z_(1)-argz_(2)|=(pi)/(2)`

Answer

A, D

Explanation

`|z_(1)^(2) -z_(2)^(2)| = |barz_(1)^(2) + barz_(1)^(2) - 2barz_(1)barz_(2)|`
or `|z_(1)-z_(2)||z_(1) + z_(2)|= |barz_(1) - barz_(2)|^(2)`
or `|z_(1) -z_(2)|= |barz _(1)= barz_(2)|`
`|z_(1)+z_(2)| = |z_(1)-z_(2)|`
`|(z_(1))/(z_(2))+1|=|(z_(1))/(z_(2))-1|` `rArr (z_(1))/(z_(2))` lies on `bot` bisector of 1 and `-1`
`rArr (z_(1))/(z_(2))` lies on imaginary axis
`rArr (z_(1))/(z_(2))` is purely imaginary
` rArr arg ((z_(1)/(z_(2))) = pm (pi)/(2))`
` |arg (z_(1)) - arg(z_(2)))|=(pi)/(2)`