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Paragraph:
Let $A, B, C$ be three sets of complex numbers as defined below.
$A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}$.Question:
Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $|w-2-i| < 3$. Then, $|z|-|w|+3$ lies between
Options:
Let $A, B, C$ be three sets of complex numbers as defined below.
$A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ; C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}$.Question:
Let $z$ be any point in $A \cap B \cap C$ and let $w$ be any point satisfying $|w-2-i| < 3$. Then, $|z|-|w|+3$ lies between
Solution:
2023 Upvotes
Verified Answer
The correct answer is:
$-3$ and 9
$-3$ and 9
$$
\begin{array}{ll}
& |w-(2+i)| < 3 \\
\Rightarrow & \quad|| w|-| 2+i|| < 3 \\
\Rightarrow & -3+\sqrt{5} < |w| < 3+\sqrt{5} \\
\Rightarrow & -3-\sqrt{5} < -|w| < 3-\sqrt{5}
\end{array}
$$
Also, $|z-(2+i)|=3$
$$
\begin{aligned}
\Rightarrow \quad-3+\sqrt{5} & \leq|z| \leq 3+\sqrt{5} \\
-3 & < |z|-|w|+3 < 9
\end{aligned}
$$
\begin{array}{ll}
& |w-(2+i)| < 3 \\
\Rightarrow & \quad|| w|-| 2+i|| < 3 \\
\Rightarrow & -3+\sqrt{5} < |w| < 3+\sqrt{5} \\
\Rightarrow & -3-\sqrt{5} < -|w| < 3-\sqrt{5}
\end{array}
$$
Also, $|z-(2+i)|=3$
$$
\begin{aligned}
\Rightarrow \quad-3+\sqrt{5} & \leq|z| \leq 3+\sqrt{5} \\
-3 & < |z|-|w|+3 < 9
\end{aligned}
$$
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