Let z = x + iy
(z + 1)/(z + i) = (x + 1 + iy) / (x + i(y + 1))
= (x + 1 + iy)(x - i(y + 1)) / (x² + (y + 1)²)
= (x(x + 1) - i(x + 1)(y + 1) + ixy - i²y(y + 1)) / (x² + (y + 1)²)
= (x² + x + y² + y + i(- xy - x - y - 1 + xy)) / (x² + (y + 1)²)
= (x² + x + y² + y + i(- x - y - 1)) / (x² + (y + 1)²)
lm[(z + 1)/(z + i)] = Re[(z + 1)/(z + i)]
=> (x² + x + y² + y) / (x² + (y + 1)²) = (- x - y - 1) / (x² + (y + 1)²)
x² + x + y² + y = - x - y - 1
x² + 2x + y² + 2y + 1 = 0
x² + 2x + 1 + y² + 2y + 1 = 1
(x + 1)² + (y + 1)² = 1
Yeah, unless my working is wrong, I got your answer.
And looking more closely, I think I know where the mistake occured.
It used (x² + x + y² + y - i(x + y + 1)) / (x² + (y + 1)²) and ignored the negative on the i when it came to finding real and imaginary parts which ended up with:
x² + x + y² + y = x + y + 1
=> x² + y² = 1