https://r2.erweima.ai/mathbot/image/1726594481370kdmw0w4c.jpg

To solve the problem in the image, let's go step by step.

### Key details from the problem:
1. **The point A** is represented as a complex number \( z \) on the Argand-Gauss plane.
2. The point A forms an angle of \( 135^\circ \) with the positive x-axis, as shown in the figure.
3. The distance from point A to the origin O is given as 4.

We need to calculate the **difference between \( z \) and its conjugate**, and find which of the provided options matches that difference.

### Step 1: Express \( z \) in polar or Cartesian form
- Since \( z \) is represented by point A, and the angle is \( 135^\circ \), we can express \( z \) using its polar coordinates. The modulus of \( z \), which is the distance from the origin to point A, is 4. The argument (angle with the positive x-axis) is \( 135^\circ \).
  
- The polar form of \( z \) is given by:
  \[
  z = r(\cos \theta + i \sin \theta)
  \]
  where \( r = 4 \) and \( \theta = 135^\circ \).

- Using trigonometric values for \( \theta = 135^\circ \):
  \[
  \cos 135^\circ = -\frac{\sqrt{2}}{2}, \quad \sin 135^\circ = \frac{\sqrt{2}}{2}
  \]

- Substituting these into the polar form:
  \[
  z = 4 \left( -\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2} \right) = -2\sqrt{2} + 2i\sqrt{2}
  \]

### Step 2: Find the conjugate of \( z \)
The conjugate of \( z \), denoted as \( \overline{z} \), is obtained by changing the sign of the imaginary part:
\[
\overline{z} = -2\sqrt{2} - 2i\sqrt{2}
\]

### Step 3: Calculate the difference \( z - \overline{z} \)
Now, we calculate the difference between \( z \) and its conjugate:
\[
z - \overline{z} = (-2\sqrt{2} + 2i\sqrt{2}) - (-2\sqrt{2} - 2i\sqrt{2})
\]
Simplifying:
\[
z - \overline{z} = (-2\sqrt{2} + 2i\sqrt{2}) + (2\sqrt{2} + 2i\sqrt{2}) = 4i\sqrt{2}
\]

### Step 4: Choose the correct option
The correct answer is:
\[
\boxed{4i\sqrt{2}}
\]

This matches the first option in the list.

---

Would you like any further details or explanations on this problem? Here are a few related questions to deepen your understanding:

1. How is the conjugate of a complex number defined in the general case?
2. What is the geometric interpretation of the conjugate of a complex number on the Argand plane?
3. How can you convert from polar to Cartesian form for any complex number?
4. How would the difference \( z - \overline{z} \) change if the angle of \( z \) were different?
5. How can you use the properties of complex numbers to solve problems involving distance in the complex plane?

### Tip:
When working with complex numbers in polar form, always remember that the modulus represents the distance from the origin, and the argument (angle) determines the direction on the Argand plane.

Na figura abaixo, o ponto A é o afixo de um número complexo z no plano de Argand-Gauss. Se a distância do ponto A até a origem O é 4, então a diferença entre z e o seu conjugado é igual a

Explore the difference between a complex number and its conjugate using polar coordinates on the Argand-Gauss plane. This problem involves converting polar to Cartesian form, finding the conjugate, and calculating the difference between the complex number and its conjugate. Suitable for students in Grades 10-12.

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