Let a complex number be


Let a complex number be $w=1-\sqrt{3} i$. Let another complex number $\mathrm{z}$ be such that $|z w|=1$ and $\arg (z)-\arg (w)=\frac{\pi}{2}$. Then the area of the triangle with vertices origin, $\mathrm{z}$ and $\mathrm{w}$ is equal to:


  1. (1) 4

  2. (2) $\frac{1}{2}$

  3. (3) $\frac{1}{4}$

  4. (4) 2

Correct Option: , 2


$\mathrm{w}=1-\sqrt{3} \cdot i \Rightarrow \mid \mathrm{wl}=2$

Now, $|z|=\frac{1}{|w|} \Rightarrow|z|=\frac{1}{2}$

and $\operatorname{amp}(\mathrm{z})=\frac{\pi}{2}+\mathrm{amp}(\mathrm{w})$

$\Rightarrow$ Area of triangle $=\frac{1}{2} \cdot$ OP. OQ

$=\frac{1}{2} \cdot 2 \cdot \frac{1}{2}=\frac{1}{2}$

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