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Question

If for complex numbers $z_{1}$ and $z_{2}, a r g (z_{1}) - a r g (z_{2}) = 0$ , then $| z_{1} - z_{2} |$ is equal to

A

| z_{1} | + | z_{2} |

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B

| z_{1} | - | z_{2} |

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C

| | z_{1} | - | z_{2} | |

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D

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Solution

The correct option is C $| | z_{1} | - | z_{2} | |$
Let
$z_{1} = r_{1} (c o s θ + i s i n θ) = r_{1} e^{i θ}$
$z_{2} = r_{2} (c o s θ + i s i n θ) = r_{2} e^{i θ}$
$z_{1} - z_{2} = (r_{1} - r_{2}) (e^{i θ})$
$| z_{1} - z_{2} | = | r_{1} - r_{2} |$
Now $r_{1} = | z_{1} |$ and $r_{2} = | z_{2} |$
$| z_{1} - z_{2} | = | | z_{1} | - | z_{2} | |$

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