If `z_1`is a complex number other than -1 such that `|z_1|=1 a n d z_2 – InterviewSolution

InterviewSolution

1.	If `z_1`is a complex number other than -1 such that `\|z_1\|=1 a n d z_2=(z_1-1)/(z_1+1)`, then show that the real parts of `z_2`is zero.
Answer» Let `z_(1) = x_(1) + iy_(1)`. Then, `\|z\|=1 rArr \|z_(1)\|= 1 rArr \|z_(1)\|^(2) = 2 rArr x_(1)^(2) + y_(1)^(2) = 1`. `z_(2)=(z_(1)-1)/(z_(1)+1)=((x_(1)+iy_(1))-1)/((x_(1)+iy_(1))+1)=((x_(1)-1)+iy_(1))/((x_(1)+1)+iy_(1))xx((x_(1)+1)-iy_(1))/((x_(1)+1)-iy_(1))` `=((x_(1)^(2)+y_(1)^(2)-1)+2iy_(1))/((x_(1)+1)^(2)+y_(1)^(2))=(2iy_(1))/((x_(1)+1)^(2)+y_(1)^(2))`, which is purely imaginary. `" "[because \|z_(1)\|^(2)=1]`

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