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

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### 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{

Consider the graph with rule |z - i| = 1 where z ∈ C. a. Write this rule in cartesian form. b. Find the points of intersection of the graphs with rules |z - i| = 1 and |z - 1| = 1 in cartesian form. c. Sketch and label the graphs with rules |z - i| = 1 and |z - 1| = 1 on the argand diagram below.

Explore the problem of expressing the complex number rule |z - i| = 1 in Cartesian form, finding the intersection points of circles defined by |z - i| = 1 and |z - 1| = 1, and sketching the graphs on an Argand diagram. This exercise involves complex numbers, Cartesian coordinates, and is suitable for Grades 11-12.

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