Question:

Step 1: Identify the mathematical property and reference point. The solution describes a property related to a complex number, specifically its argument, with respect to a fixed reference point in the complex plane. This reference point is given as $(1,1)$. Step 2: Define the argument for the first condition. For points where the real part $x$ is less than or equal to $2$, the argument of the complex number relative to point $(1,1)$ is specified as $\frac{3\pi}{4}$. $$ \text{Argument} = \frac{3\pi}{4} \quad \text{for } x \leq 2 $$ Step 3: Define the argument for the second condition. For points where the real part $x$ is greater than $2$, the argument of the complex number relative to point $(1,1)$ is specified as $-\frac{\pi}{4}$. $$ \text{Argument} = -\frac{\pi}{4} \quad \text{for } x > 2 $$ Step 4: Summarize the geometric interpretation. Combining these two conditions, the described argument represents a locus of points. Since the argument is constant for specific regions, this locus consists of two rays. Both rays originate from the reference point $(1,1)$. One ray extends for $x \leq 2$ at an angle of $\frac{3\pi}{4}$, and the other ray extends for $x > 2$ at an angle of $-\frac{\pi}{4}$.