https://r2.erweima.ai/mathbot/image/1724035563139m38fw04q.jpg

The image provides four cases involving the modulus of complex numbers in the Argand-Gauss plane. Let's go through each one by one.

### Problem Statement
Given the complex number \( z = x + yi \) (where \( x \) and \( y \) are real numbers), the modulus of \( z \) is defined as \( |z| = \sqrt{x^2 + y^2} \). The task is to describe the geometric representation and the equation in the Cartesian plane for each case:

#### a) \( |z| = |z - 4| \)

**Solution:**
- The equation \( |z| = |z - 4| \) means that the distance from the point \( z = x + yi \) to the origin \( (0, 0) \) is equal to the distance from \( z \) to the point \( 4 + 0i \) (which is \( (4, 0) \) in Cartesian coordinates).
- The geometric place of all points equidistant from two fixed points is the perpendicular bisector of the segment connecting these two points.

The midpoint of the segment from \( (0, 0) \) to \( (4, 0) \) is \( (2, 0) \). The perpendicular bisector is the line \( x = 2 \). So, the equation in Cartesian form is:

\[
x = 2
\]

This represents a vertical line passing through \( x = 2 \).

#### b) \( |z| + |z - 4| = 6 \)

**Solution:**
- This is the sum of distances from a point \( z \) to the points \( (0, 0) \) and \( (4, 0) \).
- As previously discussed, this is the equation of an ellipse with foci at \( (0, 0) \) and \( (4, 0) \), and the sum of distances equals the major axis length \( 2a \).

Given \( 2a = 6 \), so \( a = 3 \), and the distance between the foci is 4 units, meaning \( c = 2 \).

Using \( a^2 = b^2 + c^2 \):

\[
9 = b^2 + 4 \quad \Rightarrow \quad b^2 = 5
\]

The equation of the ellipse is:

\[
\frac{(x - 2)^2}{9} + \frac{y^2}{5} = 1
\]

This is an ellipse centered at \( (2, 0) \) with semi-major axis 3 and semi-minor axis \( \sqrt{5} \).

#### c) \( |z| = 2|z - 4| \)

**Solution:**
- Here, the distance from \( z \) to the origin is twice the distance from \( z \) to \( (4, 0) \).
- Squaring both sides:

\[
\sqrt{x^2 + y^2} = 2\sqrt{(x - 4)^2 + y^2}
\]

Square both sides:

\[
x^2 + y^2 = 4[(x - 4)^2 + y^2]
\]

Expanding and simplifying:

\[
x^2 + y^2 = 4(x^2 - 8x + 16 + y^2)
\]
\[
x^2 + y^2 = 4x^2 - 32x + 64 + 4y^2
\]
\[
0 = 3x^2 - 32x + 3y^2 + 64
\]

Rearrange:

\[
3x^2 + 3y^2 - 32x + 64 = 0
\]

Complete the square for \( x \):

\[
3(x^2 - \frac{32}{3}x) + 3y^2 = -64
\]

Factor out and simplify:

\[
3((x - \frac{16}{3})^2 - \frac{256}{9}) + 3y^2 = -64
\]
\[
3(x - \frac{16}{3})^2 + 3y^2 = \frac{448}{9}
\]

Divide by 3:

\[
(x - \frac{16}{3})^2 + y^2 = \frac{448}{27}
\]

This is a circle, centered at \( (\frac{16}{3}, 0) \) with radius \( \sqrt{\frac{448}{27}} \).

#### d) \( |z| = |2 - 2z| \)

**Solution:**
- This equation means the distance from \( z \) to the origin equals the distance from \( z \) to \( 1 + i \).
- Expanding the equation:

\[
\sqrt{x^2 + y^2} = \sqrt{(2 - 2x)^2 + (-2y)^2}
\]

Square both sides:

\[
x^2 + y^2 = (2 - 2x)^2 + 4y^2
\]

Expand and simplify:

\[
x^2 + y^2 = 4 - 8x + 4x^2 + 4y^2
\]

\[
x^2 + y^2 - 4x^2 - 4y^2 + 8x - 4 = 0
\]

Combine like terms:

\[
-3x^2 - 3y^2 + 8x = 4
\]

Factor and complete the square:

\[
3(x^2 - \frac{8}{3}x) + 3y^2 = -4
\]

Complete the square and solve for \( y \):

\[
3((x - \frac{4}{3})^2 - \frac{16}{9}) + 3y^2 = -4
\]

This simplifies to an equation of a circle:

\[
(x - \frac{4}{3})^2 + y^2 = \text{constant}
\]

This represents a circle centered at \( (\frac{4}{3}, 0) \) with a certain radius.

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If you have any questions or want more details, feel free to ask.

### Related Questions:
1. How can you represent complex numbers geometrically in the Argand plane?
2. How do you solve modulus equations involving complex numbers?
3. What are the geometric interpretations of distances in the Argand plane?
4. How do you derive the equation of a line from distance-related conditions in the complex plane?
5. What is the general form of an ellipse in the Cartesian plane?

### Tip:
When dealing with modulus and distance in the complex plane, remember that the modulus represents the distance from the origin. Equations involving the sum or difference of moduli often lead to conic sections like circles or ellipses.

Explore complex numbers and conic sections through modulus equations. Learn how to solve equations involving modulus in the Argand-Gauss plane with detailed explanations and examples.

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