Question

Interpret the following equations geometrically on the Argand plane

$\frac{\pi }{4}<$ arg $\left(z\right)<\frac{\pi }{3}$

• A. Region bounded by $y=x$, $y=\frac{x}{\sqrt{3}}$
• B. Region bounded by $y=x$, $y=\sqrt{3}x$
• C. Region bounded by $y=x$, $y=\sqrt{2}x$
• D. Region bounded by $y=x$, $y=2\sqrt{3}x$

Answer 1

Answer: (B)

Region bounded by $y=x$, $y=\sqrt{3}x$

We have $\frac{\pi }{4}<$ arg $\left(z\right)<\frac{\pi }{3}$
Let $z=x+iy$
$\therefore$ arg $\left(z\right)={\tan }^{-1}\left(\frac{y}{x}\right)$
The given inequality can be written as
$\frac{\pi }{4}<{\tan }^{-1}\left(\frac{y}{x}\right)<\frac{\pi }{3}$

$\Rightarrow \tan \frac{\pi }{4}<\frac{y}{x}<\tan \frac{\pi }{3}$

$\Rightarrow 1<\frac{y}{x}<\sqrt{3}$
$\Rightarrow x<y<\sqrt{3}x$
This inequality represents the region between the lines
$y=x$ and $y=\sqrt{3}x$

Hence option $B$ is correct