The problem involves determining the geometric locus of complex numbers $z$ that satisfy given conditions for the arguments of specific complex transformations. Here's a breakdown:

Step 1: Understanding the Problem

• The argument of a complex number $z = x + yi$ is defined as the angle $\theta$ that the vector from the origin to $z$ makes with the positive real axis in the complex plane.

• The conditions in this problem require determining the geometric place where the arguments (angles) of the transformed versions of $z$ are constant. Specifically, the argument conditions are:

  • $\text{Arg}(z) = \frac{3\pi}{4}$
  • $\text{Arg}(z - 2 + 2i) = \frac{3\pi}{4}$
  • $\text{Arg}(z + 2 - 2i) = \frac{3\pi}{4}$
  • $\text{Arg}(z - 2 + 2i) = \frac{\pi}{4}$

Step 2: Interpretation of Each Condition

1. $\text{Arg}(z) = \frac{3\pi}{4}$:

   • The argument $\frac{3\pi}{4}$ corresponds to a line making a $135^\circ$ angle (or $\frac{3\pi}{4}$) with the positive real axis.
   • This is the line passing through the origin and having a slope of $-1$.

2. $\text{Arg}(z - 2 + 2i) = \frac{3\pi}{4}$:

   • The term $z - (2 - 2i)$ represents a translation of $z$ by $(2, -2)$.
   • The line $\text{Arg}(z - 2 + 2i) = \frac{3\pi}{4}$ is parallel to the line $\text{Arg}(z) = \frac{3\pi}{4}$ but shifted such that it passes through $(2, -2)$.

3. $\text{Arg}(z + 2 - 2i) = \frac{3\pi}{4}$:

   • Here $z + (2 - 2i)$ translates $z$ by $(-2, 2)$.
   • The line is again parallel to the original line $\text{Arg}(z) = \frac{3\pi}{4}$ but shifted to pass through $(-2, 2)$.

4. $\text{Arg}(z - 2 + 2i) = \frac{\pi}{4}$:

   • This line corresponds to an angle of $\frac{\pi}{4}$ or $45^\circ$ with the positive real axis.
   • The line passes through $(2, -2)$ and has a slope of $1$.

Step 3: Final Observations

• Each condition describes a straight line in the complex plane:

  • Lines with $\text{Arg} = \frac{3\pi}{4}$ have a slope of $-1$.
  • Lines with $\text{Arg} = \frac{\pi}{4}$ have a slope of $1$.

• The shifts $(2, -2)$ and $(-2, 2)$ correspond to translations of the original lines.

Thus, the problem defines geometric loci that are straight lines shifted to specific points.

Do you need a visual explanation or step-by-step verification of each case? Let me know!

Related Questions to Explore:

1. What is the geometric interpretation of the argument of a complex number?
2. How do translations affect the argument of a complex number?
3. What is the slope of a line corresponding to a given argument $\theta$?
4. How can complex number transformations be represented geometrically?
5. How do you derive the equation of a line from its argument and a point?

Tip: The argument of a complex number $z = x + yi$ can be found using $\tan^{-1}(y/x)$, keeping track of the quadrant where $z$ lies.