Question:
The region represented by
$\{z=x+$ iy $\in C:|z|-\operatorname{Re}(z) \leq 1\}$ is also given by the inequality :
Correct Option: , 4
Solution:
$\mathrm{z}=\mathrm{x}+\mathrm{iy}$
$|\mathrm{z}|-\mathrm{ke}(\mathrm{z}) \leq 1$
$\Rightarrow \sqrt{x^{2}+y^{2}}-x \leq 1$
$\Rightarrow \sqrt{x^{2}+y^{2}} \leq 1+x$
$\Rightarrow x^{2}+y^{2} \leq 1+2 x+x^{2}$
$\Rightarrow y^{2} \leq 2 x+1$
$\Rightarrow y^{2} \leq 2\left(x+\frac{1}{2}\right)$