The region represented by {z=x+iy in mathbb{C | vedclass

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(D) Given the inequality $|z|-\operatorname{Re}(z) \leq 1$,where $z = x+iy$.Substituting the values,we get $\sqrt{x^{2}+y^{2}} - x \leq 1$.Rearranging

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

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(D) Given the inequality $|z|-\operatorname{Re}(z) \leq 1$,where $z = x+iy$.Substituting the values,we get $\sqrt{x^{2}+y^{2}} - x \leq 1$.Rearranging the terms,we have $\sqrt{x^{2}+y^{2}} \leq 1+x$.Since the square root is non-negative,$1+x$ must be $\geq 0$,i.e.,$x \geq -1$.Squaring both sides,we get $x^{2}+y^{2} \leq (1+x)^{2}$.Expanding the right side,$x^{2}+y^{2} \leq 1+2x+x^{2}$.Subtracting $x^{2}$ from both sides,we get $y^{2} \leq 2x+1$.Factoring out $2$,we obtain $y^{2} \leq 2\left(x+\frac{1}{2}\right)$.

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

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