The equation arg((z - 1)/(z + 1)) = (π)/(4) represents a circle...

We have,

$arg\left(\dfrac{z - 1}{z + 1}\right) = \dfrac{\pi}{4}$

$arg\left(\dfrac{{z}_{0} - 1}{{z}_{0} + 1}\right) = \dfrac{\pi}{2}$

$⇒ \left(\dfrac{{z}_{0} - 1}{{z}_{0} + 1}\right) = {e}^{i\dfrac{\pi}{2}}$

$⇒ \dfrac{{z}_{0} - 1}{{z}_{0} + 1} = i$

$⇒ {z}_{0} = \left(\dfrac{1 + i}{1 - i}\right)$

$⇒ {z}_{0} = \dfrac{( 1 + i{)}^{2}}{2}$

$⇒ {z}_{0} = i$

$⇒$ Centre \left(0, 1\right)$</p><p>Radius is the distance between${z}_{0} and \left(- 1, 0\right)

$= \sqrt{{\left(- 1 - 0\right)}^{2} + {\left(0 - 1\right)}^{2}}$

$= \sqrt{2}units$