Identify and sketch subsets of the complex plane determined by straight lines and circles, e.g. |𝑧 − 3𝑖| < 4, 𝜋 4 ≤ Arg(𝑧) ≤ 3𝜋 4, Re(𝑧) > Im(𝑧) and |𝑧 − 1| = 2|𝑧 − 𝑖|.

A ray is drawn from the point \((-2, 1)\) at an angle of \(\frac{2\pi}{3}\) to the positive real axis direction. What is the equation of this subset in the form \(\text{Arg}(z - z_0) = \theta\)?

Consider the subset of the complex plane defined by \(|z - 2 - i| = 2|z + 1 - 2i|\) for \(z \in \mathbb{C}\). (a) By letting \(z = x + yi\) where \(x, y \in \mathbb{R}\), show that this equation can be written in the form \((x - h)^2 + (y - k)^2 = r^2\) and hence identify the geometric shape. (1 mark) (b) Determine the coordinates of the centre of this shape. (1 mark) (c) Calculate the radius of this shape, giving your answer in exact form. (1 mark)

Which statement correctly describes the subset of the complex plane represented by \(\text{Arg}(z - 2 - i) = \frac{2\pi}{3}\)?