To transform the given point set $G = \{ z \in \mathbb{C} : 2 \cdot \operatorname{Re}(z) - 4 \cdot \operatorname{Im}(z) = 1 \}$ into the form $G = \{ z \in \mathbb{C} : b \cdot z + \overline{b} \cdot \overline{z} = d \}$, let's proceed with the following steps.

Step 1: Expressing $z$ in terms of its real and imaginary components

Let $z = x + iy$, where:

• $x = \operatorname{Re}(z)$ (the real part of $z$)
• $y = \operatorname{Im}(z)$ (the imaginary part of $z$)

Thus, the equation $2 \cdot \operatorname{Re}(z) - 4 \cdot \operatorname{Im}(z) = 1$ can be rewritten in terms of $x$ and $y$ as: $2x - 4y = 1$

Step 2: Rewriting the equation in terms of $z$ and $\overline{z}$

We know that: $z = x + iy \quad \text{and} \quad \overline{z} = x - iy$

From this, we can express $x$ and $y$ as: $x = \frac{z + \overline{z}}{2} \quad \text{and} \quad y = \frac{z - \overline{z}}{2i}$

Step 3: Substitute $x$ and $y$ in the original equation

Substitute $x = \frac{z + \overline{z}}{2}$ and $y = \frac{z - \overline{z}}{2i}$ into $2x - 4y = 1$: $2 \cdot \frac{z + \overline{z}}{2} - 4 \cdot \frac{z - \overline{z}}{2i} = 1$

Simplifying each term separately:

• The first term becomes $z + \overline{z}$.
• The second term becomes $-2i \cdot \frac{z - \overline{z}}{i} = 2(z - \overline{z})$.

Thus, the equation simplifies to: [ z + \overline