Question

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

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

Answer 1

The correct option is A $y^2\le 2(x+\frac{1}{2})$

$z=x+iy\in C:\left|\begin{array}{c}z\end{array}\right|-Re\left(z\right)\le 1$
$\left|\begin{array}{c}z\end{array}\right|=\sqrt{x^2+y^2}Re\left(z\right)=x$
$\left|\begin{array}{c}z\end{array}\right|-Re\left(z\right)\le 1$
$\Rightarrow \sqrt{x^2+y^2}-x\le 1$
$\Rightarrow \sqrt{x^2+y^2}\le 1+x$
$\Rightarrow x^2+y^2\le 1+x^2+2x$
$\Rightarrow y^2\le 2(x+\frac{1}{2})$