Let's go through each part of the question in detail.

Part (a)

To write the rule $|z - i| = 1$ in Cartesian form:

1. Let $z = x + yi$, where $x$ and $y$ are real numbers representing the real and imaginary parts of $z$.
2. Substitute $z = x + yi$ into the equation $|z - i| = 1$: $|(x + yi) - i| = 1$
3. Simplify the expression: $|x + (y - 1)i| = 1$
4. The modulus of a complex number $a + bi$ is ( \sqrt{