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Properties of Argument

If z1 and ...

Question

If $z_{1}$ and $z_{2}$ and two non-zero complex numbers such that $| z_{1} + z_{2} | = | z_{1} - z_{2} |$ , then arg $z_{1}$ -arg $z_{2}$ is equal to

A

- \frac{π}{2}

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B

0

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C

- π

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D

\frac{π}{2}

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Solution

The correct option is D $0$
It is given that z1 and z2 are two non-zero complex numbers such that

$| z_{1} + z_{2} | = | z_{1} | + | z_{2} |$

On squaring both sides of this equation we get

$| z_{1} |^{2} + | z_{2} |^{2} + 2 | z_{1} | | z_{2} | cos (a r g z_{1} - a r g z_{2}) = | z_{1} |^{2} + | z_{2} |^{2} + 2 | z_{1} | | z_{2} |$

This gives $2 | z_{1} | | z_{2} | cos (a r g z_{1} - a r g z_{2}) = 2 | z_{1} | | z_{2} |$

Hence, $cos (a r g z_{1} - a r g z_{2}) = 1$

Hence, $a r g (z_{1}) - a r g (z_{2}) = 0.$

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