If `|z + 1| = 1(z_(1) ne - 1 )` and `z_(2) =(z_(1)-1)/ (z_(1)-2)` , then show that the real part of `z_(2)` is zero.

Explanation

Let `z_(1) = x + iy`
`rArr |z_(1)| = sqrt(x^(2) + y ^(2)) = 1" "[:. |z_(1)|= 1, given ] ….(i)`
Now, `z_(2) = (z_(1)-1)/(z_(1)-2) = (x + iy -1)/(x + iy + 1)`
` = (x - 1 +iy)/(x + 1 + ly) =((x - 1 + iy)(x + 1 - iy))/((x +1 + iy)(x + 1 - iy))`
` = (x^(2)- 1 + ly (x + 1) - iy(x - 1) - i^(2)y^(2))/((x +1^(2))-i^(2)y^(2))`
` = (x^(2)- 1 + iyx + iy-ixy +iy + y^(2)) /((x +1^(2))+y^(2))`
` = (x^(2)+ y^(2)- 1 2iy)/((x +1^(2))+y^(2))=(1-1=2iy)/((x+1)^(2)+y^(2))" " [:x^(2) + y^(2) =1]`
Hence, the real part of `z_(2)` is zero.