Consider $z_{1}$ and $z_{2}$ are two complex numbers such that $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$
Statement – 1 : $a m p (z_{1}) - a m p (z_{2}) = 0$
Statement – 2 : The complex numbers $z_{1}$ and $z_{2}$ are collinear with origin.

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Statement – 1 is true but Statement – 2 is false.

Statement – 1 is false but Statement – 2 is true.

Both Statement – 1 and Statement – 2 are true and Statement – 2 is the correct
explanation of Statement – 1.

Both Statement – 1 and Statement – 2 are true and Statement – 2 is not the correct
explanation of Statement – 1 .

answer is A.

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Detailed Solution

Statement – 1
$∵ | z_{1} + z_{2} | = | z_{1} | + | z_{2} | \to (1)$
If $a m p (z_{1}) = θ_{1}$ and $a m p (z_{2}) = θ_{2}$ then,
${| z_{1} + z_{2} |}^{2} = {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 | z_{1} | | z_{2} | \cos (θ_{1} - θ_{2})$

$\Rightarrow {(| z_{1} | + | z_{2} |)}^{2} = {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 | z_{1} | | z_{2} | \cos (θ_{1} - θ_{2}) f r o m (1)$

$\Rightarrow {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 | z_{1} | | z_{2} | = {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 | z_{1} | | z_{2} | \cos (θ_{1} - θ_{2}) ∴ \cos (θ_{1} - θ_{2}) = 1 \Rightarrow (θ_{1} - θ_{2}) = 0 o r a m p (z_{1}) - a m p (z_{2}) = 0$