The region represented by


The region represented by $\{z=x+i y \in \mathrm{C}:|z|-\operatorname{Re}(z) \leq 1\}$ is also given by the inequality:

  1. (1) $y^{2} \geq 2(x+1)$

  2. (2) $y^{2} \leq 2\left(x+\frac{1}{2}\right)$

  3. (3) $y^{2} \leq x+\frac{1}{2}$

  4. (4) $y^{2} \geq x+1$

Correct Option: 2,


$\because|z|-\operatorname{Re}(z) \leq 1 \quad(\because z=x+i y)$

$\Rightarrow \sqrt{x^{2}+y^{2}}-x \leq 1 \Rightarrow \sqrt{x^{2}+y^{2}} \leq 1+x$

$\Rightarrow x^{2}+y^{2} \leq 1+x^{2}+2 x$

$\Rightarrow y^{2} \leq 1+2 x \Rightarrow y^{2} \leq 2\left(x+\frac{1}{2}\right)$

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