Question

The area bounder by the curves $arg$ $z=\frac{\pi }{3}$ and $arg$ $z=\frac{2\pi }{3}$ and $arg$$(z-2-2i\sqrt{3})=\pi$ in the argand plane is

• A. $4\sqrt{3}$
• B. $2\sqrt{3}$
• C. $\sqrt{3}$
• D. $3\sqrt{3}$

Answer 1

The correct option is A $4\sqrt{3}$

Let, $z=x+iy$.

Now, $arg\left(z\right)=\frac{\pi }{3}\Rightarrow$ ${\tan }^{-1}\frac{y}{x}=\frac{\pi }{3}$

or, $y=\sqrt{3}x$.......(1).

Also, $arg\left(z\right)=\frac{2\pi }{3}\Rightarrow$ $y=-\sqrt{3}x$.........(2).

Also, $arg(z-2-2i\sqrt{3})=\pi \Rightarrow {\tan }^{-1}\frac{y-2\sqrt{3}}{x-2}$ $=\pi \Rightarrow y=2\sqrt{3}$.......(3).

Now, the area bounded by the curves (1), (2) and (3)

$=2\times \Delta$OAB

$=2\times \frac{1}{2}$ $\times 2\times 2\sqrt{3}=4\sqrt{3}$