If z and w are two complex numbers such that |zw|=1 

and arg(z)-arg(w)=\frac{\pi}{2},  then : 

  • Option 1)

  • Option 2)

    z\bar w=\frac{-1+i}{\sqrt2}

  • Option 3)

    \bar z w=-i

  • Option 4)

    z\bar w=\frac{1-i}{\sqrt2}

Answers (1)

|zw|=1 and arg(z)-arg(w)=\frac{\pi}{2},

Let |z|=r   => z=re^{i\theta}

|\omega |=\frac{1}{r}    => \omega =\frac{1}{r}e^{i\phi }

arg(z)-arg(w)=\frac{\pi}{2}

\theta -\phi =\frac{\pi}{2}

\theta =\frac{\pi}{2}+\phi

z\bar{\omega }=re^{i\theta }.\frac{1}{r}e^{-i\phi }

        =re^{i(\theta -\phi)}    

        =re^{i(\frac{\pi}{2}+\phi -\phi)}

        =re^{i(\frac{\pi}{2})}

        =cos(\frac{\pi}{2})+isin(\frac{\pi}{2})

        =0+i.1

        =i

 


Option 1)

Option 2)

z\bar w=\frac{-1+i}{\sqrt2}

Option 3)

\bar z w=-i

Option 4)

z\bar w=\frac{1-i}{\sqrt2}

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