Let z=1+i and z1=(1+i barz/ barz(1-z)+(1)z). Then (12/π) arg (z1) is equal to

Question

Let z=1+i and z1=(1+i barz/ barz(1-z)+(1)z). Then (12/π) arg (z1) is equal to (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

z-1+i z1=(1+i barz/ barz(1-z)+(1)z) z1=(1+i(1 i)/(1-i)(1-1-i)+(1)1+i) =(1+i-i2/(1-i)(-i)+(1-i)2) =(2+i/-3 i-1)=(4+2 i/-3 i-1) =(-(4+2 i)(3 i-1)/(3 i)2-(1)2) textArg(z1)=(3 π/4) ∴ (12/π) arg ( z 1)=(12/π) × (3 π/4)=9

Q. Let z=1+i and z1​=zˉ(1−z)+z1​1+izˉ​. Then π12​arg(z1​) is equal to________

z−1+i z1​=zˉ(1−z)+z1​1+izˉ​ z1​=(1−i)(1−1−i)+1+i1​1+i(1i)​ =(1−i)(−i)+21−i​1+i−i2​ =−3i−12+i​=−3i−14+2i​ =(3i)2−(1)2−(4+2i)(3i−1)​ Arg(z1​)=43π​ ∴π12​arg(z1​)=π12​×43π​=9

Q. Let $z = 1 + i$ and $z_{1} = \frac{1 + i z ˉ}{z ˉ ( 1 - z ) + \frac{1}{z}}$ . Then $\frac{12}{π} ar g (z_{1})$ is equal to________