z is complex number such that ` (z-i)/(z-1)` is purely imaginary. Show that z lies on the circle whose centres is the point ` 1/2 (1+i)` and radius is ` 1/sqrt2`

Explanation

To solve the problem, we need to show that the complex number \( z \) lies on a specific circle defined by its center and radius. We start with the given condition that \( \frac{z - i}{z - 1} \) is purely imaginary.

### Step-by-step Solution:

1. **Define the complex number**:
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

2. **Set up the equation**:
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