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Let's begin by analyzing each part of the problem you've shared. We're asked to describe the sets analytically and plot them on the complex plane.

### (a) \( A = \{ z \in \mathbb{C} : |z - 1| = 2|z - i| \} \)

- This represents the locus of points \( z \) in the complex plane such that the distance from \( z \) to the point \( 1 \) is twice the distance from \( z \) to \( i \). This is the equation of an **Apollonian circle**.
  
- **Analytical description**: This is a circle. The equation represents the set of points that are at a constant ratio of distances to two fixed points, in this case, \( 1 \) and \( i \).

- **Plot**: The result will be a circle, and we can determine its center and radius through algebraic manipulation.

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### (b) \( B = \{ z \in \mathbb{C} : |z + i| + |z - i| = 4 \} \)

- This is the set of points such that the sum of the distances from \( z \) to \( -i \) and \( z \) to \( i \) is constant (equal to 4). This describes an **ellipse** where the foci are at \( -i \) and \( i \).

- **Analytical description**: This is an ellipse, with foci at \( z = -i \) and \( z = i \) and the sum of distances from these foci is 4.

- **Plot**: We can sketch an ellipse on the complex plane centered between \( -i \) and \( i \).

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### (c) \( C = \{ z \in \mathbb{C} : z\overline{z} + iz - i\overline{z} = 3 \} \)

- Here, \( \overline{z} \) is the complex conjugate of \( z \). First, let \( z = x + iy \) where \( x, y \) are real numbers. Substituting this into the equation will give us a real equation.

- **Analytical description**: This will lead to either a line or a circle once simplified.

- **Plot**: After simplifying, we will know whether it’s a circle, a line, or some other geometric shape.

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### (d) \( D = \{ z \in \mathbb{C} : | \text{Arg}(z) - \text{Arg}(i)| < \frac{\pi}{6} \} \)

- This set consists of points where the argument (angle) of \( z \) lies within \( \frac{\pi}{6} \) radians of the argument of \( i \), which is \( \frac{\pi}{2} \).

- **Analytical description**: This is a **wedge-shaped region** in the complex plane, symmetric about the positive imaginary axis, with an angular width of \( \frac{\pi}{3} \).

- **Plot**: We will plot a wedge with an angular width of \( \frac{\pi}{3} \), centered on the positive imaginary axis.

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### (e) \( E = \{ z \in \mathbb{C} : \text{Arg}(z) + \text{Arg}(z - 1) = \pi \} \)

- This set describes points where the sum of the arguments of \( z \) and \( z - 1 \) is \( \pi \). This implies that \( z \) and \( z - 1 \) are collinear with the origin.

- **Analytical description**: This is a **straight line** that passes through \( z = 1 \) and the origin.

- **Plot**: The result is a straight line in the complex plane, passing through \( 1 \) and the origin.

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Would you like to see the algebraic steps for any of these parts, or do you want to focus on specific plots or deeper details on certain sets?

Here are five related questions for further exploration:
1. How do we algebraically derive the equation for a circle from an Apollonian distance condition like in part (a)?
2. Can you explain how to find the major and minor axes of an ellipse in part (b)?
3. What would happen to the geometry in part (c) if we changed the constant on the right-hand side of the equation?
4. How can you verify that the region in part (d) is correctly bounded by the angle \( \frac{\pi}{3} \)?
5. What is the significance of the condition in part (e) that makes the points collinear?

**Tip**: When solving complex plane problems, visualize the geometry and use the modulus and argument properties to simplify the interpretation.

For each of the following sets, first describe it analytically and then draw it in the complex plane. (a) A = { z ∈ C : |z - 1| = 2|z - i| }, (b) B = { z ∈ C : |z + i| + |z - i| = 4 }, (c) C = { z ∈ C : z\overline{z} + iz - i\overline{z} = 3 }, (d) D = { z ∈ C : Arg(z) - Arg(i) < π/6 }, (e) E = { z ∈ C : Arg(z) + Arg(z - 1) = π }

Explore how to analyze and plot various geometric sets in the complex plane, involving circles, ellipses, and wedge-shaped regions. This problem covers key concepts in complex number geometry, including the Apollonian circle, ellipses, and the argument function.

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