If z is a complex number of unit modulus and argument θ , then arg 1 + z 1 + z - can be equal to given…

If z is a complex number of unit modulus and argument θ, then arg 1+z1+z- can be equal to given z≠-1
  1. θ
  2. π - θ
  3. - θ
  4. π 2 - θ

Solution

Let z=cos θ+i sin θ

Then, 1+z1+z¯=1+cos θ+i sin θ1+cos θ-i sin θ

=2 cos2θ2+i·2 sinθ2cosθ22 cos2θ2-i·2 sinθ2cosθ2

=cosθ2+i sinθ2cosθ2-i sinθ2

=cosθ2+i sinθ22

=cos θ+i sin θ

∴   arg1+z1+z¯=arg z

=θ

Asked in: JEE Main 2013 (07 Apr)

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