If z is a complex number such that (z - i)/(z - 1) is purely...

$∵ \dfrac{z - i}{z - 1}$ is purely imaginary.

$arg\left(\dfrac{z - i}{z - 1}\right) = \dfrac{\pi}{2} + n\pi; n \in I$

Geometrically, the locus of $z$ is the circle with end points of diameter is \left(1, 0\right) \text{ and } \left(0, 1\right)$and centre is \left(\dfrac{1}{2}, \dfrac{1}{2}\right)$.So the nearest point to \left(3, 3\right)$on the circle is A\left(1, 1\right)$.

Minimum distance is $AP = OP - OA$

$= \sqrt{{3}^{2} + {3}^{2}} - \sqrt{{1}^{2} + {1}^{2}}$

$= 3\sqrt{2} - \sqrt{2}$

$= 2\sqrt{2}$