Answer lies in the question

Algebra Level 4

Let $z$ and $w$ be complex numbers such that $\overline{z} + i \overline{w} = 0$ and $\text{arg}(zw) = \pi$ , then evaluate $\text{arg}(z)$ .

{ \pi }/{ 2 }

{ \pi }/{ 4 }

{ 3\pi }/{ 4 }

{ 5\pi }/{ 4 }

1 solution

Bakul Choudhary
Aug 6, 2015

Given, $\bar{z}+i\bar{w} =0$ taking conjugate of both sides, $z-iw=0$ because conjugate of sum is sum of conjugates and conjugate of product is product of conjugates. So, z=iw $arg(zw)=arg(z)+arg(w)\\ or,\quad arg(z)=\pi -arg(w)\ \ \ \ \ \ \_ \_ (1)$ and, $arg(z)= arg(i) +arg(w)\\ or, \quad arg(z) = \frac { \pi }{ 2 } + arq(w) \ \ \ \ \ \ \_ \_ (2)$ adding equations (1) and (2), $2arg(z)=\pi +\frac { \pi }{ 2 }$ therefore, $\boxed { arg(z)=\frac { 3\pi }{ 4 } }$

Answer lies in the question

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