Question

The area bounded by $|\arg \left(z\right)|\le \frac{\pi }{4}$ and $|z-1|<|z-3|$ is

• A. $1$ sq. unit
• B. $2$ sq. units
• C. $4$ sq. units
• D. $8$ sq. units

Answer 1

The correct option is C $4$ sq. units

$\arg |z|<\frac{\pi }{4}$
$\Rightarrow -\frac{\pi }{4}<\text{arg z}<\frac{\pi }{4}$
Which represents the region given in the following diagram.


$|z-1|<|z-3|\Rightarrow (x-1{)}^2+y^2<(x-3{)}^2+y^2$
$\Rightarrow x<\mathrm{2....}\left(ii\right)$


Common region of $\left(i\right)$ and $\left(ii\right)$ is shown in above figure.
$\text{Area of the shaded region}=2\times \frac{1}{2}\left(2\right)\left(2\right)=4\text{sq.units}$