To find the value of complex number

Question

If $z$ and $w$ are two non zero complex numbers such that $|zw| =1$ and $\arg z - \arg w = \pi/2$ then conjugate of $(zw)$ =?

Answer 1

Hint. Write $z$ and $w$ in polar form, $$z=re^{i\alpha}\quad\hbox{and}\quad w=se^{i\beta}\ .$$ Then

• in terms of $r,s$ we have $|zw|=\cdots$
• in terms of $\alpha,\beta$ we have ${\rm Arg}(z)-{\rm Arg}(w)=\cdots$
• and so $\overline{zw}=\cdots$

Answer 2

Let, $$z=|z|e^{i\arg z},\quad w=|w|e^{i\arg w}$$ Hence, $\overline{zw}=|zw|e^{-i(\arg z +\arg w)}=e^{-i(\pi/2+2\arg w)}=-ie^{-i2\arg w}$. Hope this helps.