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  • If z and w are two complex numbers such that |zw| = 1 and arg (z) – arg (w) = π/2, then show that z ̅w=-i.

If z and w are two complex numbers such that |zw| = 1 and arg(z) - arg (w) = \pi/2, then show that zw=-i.

Answers (1)

Let  z=\left | z \right |(\cos\theta_1 + i sin\theta_1 )  and w=\left | w \right |(\cos\theta_2 + i sin\theta_2 )

|zw|=|z||w|=1

Also arg(z)-arg(w)= \frac{\pi}{2}

\theta_1-\theta_2=\frac{\pi}{2}

\bar{z} w=|z|\left (\cos \theta _{1}-i\sin\theta _{1} \right )w

w=|w|\left (\cos \theta _{2}+i\sin\theta _{2} \right )=1

\bar{z}w=|z||w|(\cos (-\theta_1)+i sin(-\theta_1 ))(cos\theta_2+i \sin\theta_2 )

\bar{z}w=\cos [(\theta _2 - \theta _1 )+isin( \theta _2- \theta _1 )]

=\left [ \cos \left ( -\frac{\pi}{2} \right ) +i\sin\left ( -\frac{\pi}{2} \right )\right ]

=1[0-i]=-i

 

 

 

 

Posted by

infoexpert21

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