Prove that complex number can represented in this form

[gurty]

Hello

I'm having a bit of trouble with the following:
Prove that if complex number z != -1 and modulus of the z is 1, then z can be represented as z = (1 + xi) / (1 - xi), where x is a real number.

What I got so far:
z = a + bi
a*a + b*b = 1
z = sqrt(1 - b*b) + bi
Now I would guess, I have to convert that last line to the required form of z = (1 + xi) / (1 - xi), but I simply can't figure this out.

Thanks


EDIT: Solved it just a few hours after posting it

 

[Deleted member 4993]

Hello

I'm having a bit of trouble with the following:
Prove that if complex number z != -1 and modulus of the z is 1, then z can be represented as z = (1 + xi) / (1 - xi), where x is a real number.

What I got so far:
z = a + bi
a*a + b*b = 1
z = sqrt(1 - b*b) + bi
Now I would guess, I have to convert that last line to the required form of z = (1 + xi) / (1 - xi), but I simply can't figure this out.

Thanks


EDIT: Solved it just a few hours after posting it

I assume that you have shown:

\(\displaystyle x \ = \dfrac{-a \ \pm \ \sqrt{a^2+b^2}}{b} \ = \ \dfrac{-a \ \pm \ 1}{b}\)