Question

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

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

Answer 1

The correct option is A

$\frac{1}{2}$

Explanation for correct answer:

Finding the area of the triangle:

Given, $w=1-\sqrt{3i}$

$$\begin{gathered} \left|w\right|=\sqrt{1^2+{\left(\sqrt{3}\right)}^2} \\ =\sqrt{4}=2 \end{gathered}$$

$$\begin{gathered} \left|zw\right|=1 \\ \left|z\right|\left|w\right|=1 \\ \Rightarrow \left|z\right|=\frac{1}{\left|w\right|}=\frac{1}{2}\left[\because \left|w\right|=2\right] \end{gathered}$$

$\text{arg}\left(z\right)-\text{arg}\left(w\right)=\frac{\mathrm{\pi }}{2}$

Finding the area of triangle, vertices are origin, $z$ and $w$.

From the diagram, we get to know

Base =$2$

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

Area of triangle$=\frac{1}{2}\times \left(\text{base}\right)\times \left(\text{height}\right)$

$$\begin{gathered} =\frac{1}{2}\times 2\times \frac{1}{2} \\ =\frac{1}{2} \end{gathered}$$

Hence, option (A) is the correct answer