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  • z1 and z2 are two complex numbers such that |z1| = |z2| and arg (z1) + arg (z2) = π, then show that z_1=-z_(2 ).

 z1 and z2 are two complex numbers such that |z_1| = |z_2| and arg (z_1) + arg (z_2) = \pi, then show that |z_1| = -|z_2|  .

Answers (1)

Let z_1=\left |z_1 \right |\left (cos\theta _1+i\ sin\theta_1 \right ) and z_2=\left |z_2 \right |\left (cos\theta _2+i\ sin\theta_2 \right )

Given that |z_1| = |z_2| 

And   arg\left (z_1 \right )+arg\left (z_2 \right )= \pi

\theta_1+\theta_2=\pi

\theta_1=\pi-\theta_2

z_2=\left |z_2 \right |\left (cos\theta _2+i\ sin\theta_2 \right )

z_1=|z_2 |\left ( -\cos \theta_2+isin\theta_2 \right )

z_1=-|z_2 |\left ( \cos \theta_2-isin\theta_2 \right )

z_1=-|z_2 |\left ( \cos \theta_2-isin\theta_2 \right ), |z_1| = -|z_2|

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infoexpert21

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