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Given z is a complex number with modulus...

Given `z` is a complex number with modulus 1. Then the equation `[(1+i a)//(1-i a)]^4=z` has all roots real and distinct two real and two imaginary three roots two imaginary one root real and three imaginary

A

all roots real and distinct

B

two real and tw imaginary

C

three roots real and one imaginary

D

one root real and three imaginary

Text Solution

Verified by Experts

The correct Answer is:
A
`((1+ia)/(1-ia))^(4) =z`
or ` |(1+ia)/(1-ia)|^(4) = |z|`
or `|(a-i)/(a+i)|^(4) = 1`
` or |a -i|= |a + i|`
Therefore, a lies on the perpendicular bisector of iand -i. which is the real axis. Hence,all the roots are real.Obvioulsy roots are distinct.
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