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

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

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

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

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

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

$\because y\ne 0 \mathrm{a}\mathrm{n}\mathrm{d} a-b=1$

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

$\Rightarrow x=-1\pm \sqrt{1-y^2}$