Question

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

• A. 4
• B. $\frac{1}{2}$
• C. $\frac{1}{4}$
• D. 2

Answer 1

Answer: (B)

$w =1-\sqrt{3} . i \Rightarrow | w |=2$
Now, $| z |=\frac{1}{| w |} \Rightarrow | z |=\frac{1}{2}$
and $\operatorname{amp}( z )=\frac{\pi}{2}+\operatorname{amp}( w )$

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