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Modulus of a Complex Number

If z1 lies ...

Question

If $z_{1}$ lies on $| z | = 1$ and $z_{2}$ lies on $| z | = 2$ , then

A

3 \leq | z_{1} - 2 z_{2} | \leq 5

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B

1 \leq | z_{1} + z_{2} | \leq 3

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C

| z_{1} - 3 z_{2} | \geq 5

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D

| z_{1} - z_{2} | \geq 1

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Solution

The correct options are
A $1 \leq | z_{1} + z_{2} | \leq 3$
B $| z_{1} - 3 z_{2} | \geq 5$
C $3 \leq | z_{1} - 2 z_{2} | \leq 5$
D $| z_{1} - z_{2} | \geq 1$
Given that, $| z_{1} | = 1$ and $| z_{2} | = 2$
We know that $| | z | - | w | | \leq | z - w | \leq | z | + | w |$
Now, $| | z_{1} | - | 2 z_{2} | | \leq | z_{1} - 2 z_{2} | \leq | z_{1} | + | 2 z_{2} |$
$\Rightarrow | 1 - 2 (2) | \leq | z_{1} - 2 z_{2} | \leq 1 + 2 (2)$
$\Rightarrow 3 \leq | z_{1} - 2 z_{2} | \leq 5$
and $| | z_{1} | - | z_{2} | | \leq | z_{1} + z_{2} | \leq | z_{1} | + | z_{2} |$
$\Rightarrow | 1 - (2) | \leq | z_{1} + z_{2} | \leq 1 + (2)$
$\Rightarrow 1 \leq | z_{1} + z_{2} | \leq 3$
and $| z_{1} - 3 z_{2} | \geq | | z_{1} | - | 3 z_{2} | |$
$\Rightarrow | z_{1} - 3 z_{2} | \geq | 1 - 3 (2) |$
$\Rightarrow | z_{1} - 3 z_{2} | \geq 5$
and $| z_{1} - z_{2} | \geq | | z_{1} | - | z_{2} | |$
$\Rightarrow | z_{1} - z_{2} | \geq | 1 - (2) |$
$\Rightarrow | z_{1} - z_{2} | \geq 1$
Ans: A,B,C,D

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