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Modulus of a Complex Number

|Z1-3-4i|=2 a...

Question

$| Z_{1} - 3 - 4 i | = 2$ and $| Z_{2} - 3 - 4 i | = 5$ , then the minimum distance between the $Z_{1}$ and $Z_{2}$ is

A

5

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B

7

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C

3

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D

12

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Solution

The correct option is C $3$
$| Z_{1} - (3 + 4 i) | = 2$
$| Z_{1} - (3 + 4 i) | \geq | Z_{1} | - 5$
$2 \geq | Z_{1} | - 5$
$| Z_{1} | \leq 7$
$| Z_{2} - (3 + 4 i) | = 5$
$| Z_{2} - (3 + 4 i) | \geq | Z_{2} | - 5$
$5 \geq | Z_{2} | - 5$
$| Z_{2} | \leq 10$
$| Z_{1} - Z_{2} | \geq | Z_{2} | - | Z_{1} | ⟹ \geq 10 - 7 ⟹ 3$

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