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# I need help with - Complex numbers and quadratic equations - JEE Main-13

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)

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

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$|zw|=1$ 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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