Question

A rectangle is constructed in the complex plane with its sides parallel to the axes and its center is situated at the origin. If one of the vertices of the rectangle is $a+ib\sqrt{3}$, then the area of the rectangle is
A. $ab\sqrt{3}$
B. $2ab\sqrt{3}$
C. $3ab\sqrt{3}$
D. $4ab\sqrt{3}$

Answer 1

Hint: In this question, we have to find the area of the given rectangle. Here the rectangle is constructed in such a way that, it is divided into four equal parts, and each part is another rectangle. So, the length and width of a rectangular part can able to find with the given details. In such a way we can calculate the area of the rectangle.

Formula Used: The complex number $(x,y)$ is represented by $x+iy$.
If $z=x+iy\in C$, then $x$ is called the real part and $y$ is called the imaginary part of $z$. These are represented by $\operatorname{Re}(z)$ and $\operatorname{Im}(z)$ respectively.
$z=x+iy$ be a complex number such that $\left| z \right|=r$ and $\theta $ be the amplitude of $z$. So, $\cos \theta =\dfrac{x}{r},\sin \theta =\dfrac{b}{r}$
And we can write the magnitude as
 $\begin{align}
  & \left| z \right|=\left| x+iy \right| \\
 & \Rightarrow r=\sqrt{{{x}^{2}}+{{y}^{2}}} \\
\end{align}$
Thus, we can write
$z=x+iy=r\cos \theta +ir\sin \theta =r(\cos \theta +i\sin \theta )$
This is said to be the mod amplitude form or the polar form of $z$.
Where $\cos \theta +i\sin \theta $ is denoted by $cis\theta $ and the Euler’s formula is $\cos \theta +i\sin \theta ={{e}^{i\theta }}$

Complete step by step solution: Given that,
The rectangle is formed in such a way that its sides are parallel to the axes and the center of the rectangle is the origin.
The rectangle is as shown below:

Consider the given coordinate as $B=(a,b\sqrt{3})=a+ib\sqrt{3}$
Then, the other vertices at one part of the rectangle are
$O(0,0);A(a,0);C(0,b\sqrt{3})$
Then the area of that part of the rectangle $(OABC)$ is
$\begin{align}
  & Area=l\times b \\
 & \text{ }=a\times b\sqrt{3} \\
 & \text{ }=ab\sqrt{3} \\
\end{align}$
Thus, the area of the whole rectangle is
$\begin{align}
  & =4\times Area(OABC) \\
 & =4\times ab\sqrt{3} \\
 & =4ab\sqrt{3} \\
\end{align}$

Option ‘D’ is correct

Note: Here we need to remember that the rectangle has its centre at the centre of the coordinate axes. So, we can able to calculate the area of one of the four parts of the rectangle.