Question

Let z be any point in $A\cap B\cap C$ and let w be any point satisfying $|w-2-i|<3.$ Then, $\left|z\right|-\left|w\right|+3$ lies between

• A. -6 and 3
• B. -3 and 6
• C. -6 and 6
• D. -3 and 9

Answer 1

The correct option is D -3 and 9

Since $|w-(2+i)|<3$
$\Rightarrow \left|w\right|-|2+i|<3\Rightarrow -3+\sqrt{5}<\left|w\right|<3+\sqrt{5}\Rightarrow -3-\sqrt{5}<\left|w\right|<3-\sqrt{5}$
Also $|z-(2+i)|=3$
$\Rightarrow -3+\sqrt{5}\le \left|z\right|\le 3+\sqrt{5}\therefore -3<\left|z\right|-\left|w\right|+3<9$