The equation Im((iz - 2)/(z - i)) + 1 = 0, z ∈ C, z ≠ i represents...

Let $z = x + iy,$ then

$Im\left[\left(\dfrac{ix - y - 2}{x + \left(y - 1\right)i}\right)\left(\dfrac{x - \left(y - 1\right)i}{x - \left(y - 1\right)i}\right)\right] + 1 = \dfrac{\left(y - 1\right)\left(y + 2\right) + {x}^{2}}{{x}^{2} + {\left(y - 1\right)}^{2}} + 1 = 0$

$⇒ 2{x}^{2} + 2{y}^{2} - y - 1 = 0$

$⇒ {x}^{2} + {y}^{2} - \dfrac{y}{2} - \dfrac{1}{2} = 0$

Center $( 0, \dfrac{1}{4} )$

Radius $= \sqrt{{g}^{2} + {f}^{2} - c}$

Radius$= \sqrt{\dfrac{1}{16} + \dfrac{1}{2}} = \sqrt{\dfrac{9}{16}} = \dfrac{3}{4}$