Let a, b, x and y be real numbers such that a - b = 1 and y ≠ 0 ....

$Im\left(\dfrac{az + b}{z + 1}\right) = y$ and $z = x + iy$

$∴ Im\left(\dfrac{a\left(x + iy\right) + b}{x + iy + 1}\right) = y$

multiply and divide by $x + 1 - iy$ and equate imaginary part

$⇒ - y\left(ax + b\right) + ay\left(x + 1\right) = y\left({\left(x + 1\right)}^{2} + {y}^{2}\right)$

$⇒ \left(a - b\right)y = y\left({\left(x + 1\right)}^{2} + {y}^{2}\right)$

$∵ y \ne 0anda - b = 1$

$⇒ {\left(x + 1\right)}^{2} + {y}^{2} = 1$

$⇒ x = - 1 \pm \sqrt{1 - {y}^{2}}$