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Modulus of a Complex Number

For two compl...

Question

For two complex numbers $z_{1}, z_{2}$ the relation $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$ holds, if

A

a r g (z_{1}) = a r g (z_{2})

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B

a r g (z_{1}) = a r g (z_{2}) = \frac{π}{2}

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C

z_{1} z_{2} = 1

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D

| z_{1} | = |_{1} z_{2} |

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Solution

The correct option is A $a r g (z_{1}) = a r g (z_{2})$
Since, $| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$
$\Rightarrow {| z_{1} + z_{2} |}^{2} = {(| z_{1} | + | z_{2} |)}^{2}$
$\Rightarrow {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 | z_{1} | | z_{2} | cos (θ_{1} - θ_{2})$
$= {| z_{1} |}^{2} + {| z_{2} |}^{2} + 2 | z_{1} | | z_{2} |$
$\Rightarrow cos (θ_{1} - θ_{2}) = 1 = cos 0^{\circ}$
$\Rightarrow θ_{1} - θ_{2} = 0 \Rightarrow θ_{1} = θ_{2}$
$\Rightarrow a r g (z_{1}) = a r g (z_{2})$

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Similar questions

z_{1}, z_{2}

are complex numbers then the correct match List- I to List II is:

List - I	List - II
A) arg z_1z_2	1) $a r g z_{1} - a r g z_{2}$
B) $a r g z_{1}_{2}$	2) $a r g z_{1} - a r g z_{2} = \frac{π}{2}$
C) $\| z_{1} + z_{2} \| = \| z_{1} - z_{2} \|$	3) $a r g z_{1} = a r g z_{2}$
D) $\| z_{1} + z_{2} \|^{2}$	4) $a r g z_{1} + a r g z_{2}$
E) $\| z_{1} + z_{2} \| = \| z_{1} \| + \| z_{2} \|$	5) $r_{1}^{2} + r_{2}^{2} + 2 r_{1} r_{2} c o s (θ_{1} - θ_{2})$

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