Find the locus of a complex number `z= x+ yi` satisfying the relation arg `(z-a)=(pi)/(4), a in R`
Illustrate the locus of z in the Argand plane

Answer

To find the locus of the complex number \( z = x + yi \) satisfying the relation \( \arg(z - a) = \frac{\pi}{4} \), where \( a \) is a real number, we can follow these steps:

### Step 1: Express the complex number
We start by expressing the complex number \( z \):
\[ z = x + yi \]
where \( x \) and \( y \) are the real and imaginary parts of \( z \), respectively.

### Step 2: Substitute into the argument condition
We need to calculate \( z - a \):
\[ z - a = (x - a) + yi \]

### Step 3: Set up the argument condition
The argument of a complex number \( u + vi \) is given by \( \arg(u + vi) = \tan^{-1}\left(\frac{v}{u}\right) \). Therefore, we have:
\[ \arg(z - a) = \arg((x - a) + yi) = \tan^{-1}\left(\frac{y}{x - a}\right) \]
According to the problem, this is equal to \( \frac{\pi}{4} \):
\[ \tan^{-1}\left(\frac{y}{x - a}\right) = \frac{\pi}{4} \]

### Step 4: Apply the tangent function
Taking the tangent of both sides gives us:
\[ \frac{y}{x - a} = \tan\left(\frac{\pi}{4}\right) \]
Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), we can write:
\[ \frac{y}{x - a} = 1 \]

### Step 5: Rearranging the equation
From the equation above, we can rearrange it to find a relationship between \( x \) and \( y \):
\[ y = x - a \]

### Step 6: Identify the locus
This equation \( y = x - a \) represents a family of straight lines in the Argand plane (complex plane) with a slope of 1. The value of \( a \) shifts the line vertically. Thus, as \( a \) varies over all real numbers, the locus of \( z \) is a family of lines, each parallel to the line \( y = x \).

### Conclusion
The locus of the complex number \( z \) satisfying the given condition is a set of lines in the Argand plane, each with a slope of 1, and shifted vertically by \( a \).

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