If `z` is a complex number such that `|z|=1,` prove that `(z-1)/(z+1)` is purely imaginary, what will by your conclusion if `z=1?`

Answer

N/a

Explanation

Let `z=x+iy`
`rArr" "|z|=|x+iy|`
`rArr" "1=sqrt(x^(2)+y ^(2))`
`rArr" "x^(2)+ y^(2 )=1`
Now,
`" "(z-1)/(z+1) =(x+iy-1)/(x+iy+1)`
`" "=((x-1)+iy)/((x+1)+iy)xx((x+1)-iy)/((x+1)-iy)`
`" "=(x^(2)+x-ixy-x-1+iy+ixy+iy-i^(2)y^(2))/((x+1)^(2) -i^(2)y^(2))`
` " "=(x^(2)+y^(2)-1+2iy)/(x^(2)+1+2x +y^(2))`
`" "=(1-1+2iy)/(1+1+2x)=i*(y)/( 1+x)`
which is a pure imaginary number. Hence Proved.