Question

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

• A. $y^2\le 2\left(x+\frac{1}{2}\right)$
• B. $y^2\le \left(x+\frac{1}{2}\right)$
• C. $y^2\le 2\left(x+1\right)$
• D. $y^2\le \left(x+1\right)$

Answer 1

The correct option is A

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

Explanation for the correct answer:

Compute the inequality:

$z=x+iy$

$Re\left(z\right)=x$

$\left|z\right|=\sqrt{x^2+y^2}$

According to given inequality,

$\left|z\right|-Re\left(z\right)\le 1$

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

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

$\Rightarrow$ $x^2+y^2\le x^2+2x+1$

$\Rightarrow$ $y^2\le 2x+1$

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

Hence, option $\left(A\right)$ is the correct answer.