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Summation of Determinant

120. If |z1+z...

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120. If |z1+z2|=|z1|+|z2|,then proved that argument (z1)=argument (z2)

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

then the difference between the arguments of

z_{1}

z_{2}

z_{1}, z_{2} \in C_{1}

then prove that :

| z_{1} - z_{2} | \geq | z_{z} | - | z_{2} |

z_{1}

z_{2}

are two complex numbers, then prove that

| z_{1} | + | z_{2} | = ∣ ∣ ∣ \frac{z_{1} + z_{2}}{2} + \sqrt{z_{1} z_{2}} ∣ ∣ ∣ + ∣ ∣ ∣ \frac{z_{1} + z_{2}}{2} - \sqrt{z_{1} z_{2}} ∣ ∣ ∣

Q. Consider the complex number

z_{1}

z_{2}

satisfying the relation

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

, then the possible difference between the argument of

z_{1}

z_{2}

Q. (a) For any two non-zero complex numbers

z_{1}

z_{2}

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

, then prove that

a r g z_{1} - a r g z_{2}

is zero.
(b) Prove the above result if we have

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

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