If `z= x+iy` and the amplitude of `(z-2-3i)` is `(pi)/(4)`. Find the relation between x and y.

Answer

To solve the problem, we start with the given information:

Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

We are given that the amplitude (or argument) of \( z - 2 - 3i \) is \( \frac{\pi}{4} \).

### Step 1: Express \( z - 2 - 3i \)
We can rewrite \( z - 2 - 3i \) as:
\[ z - 2 - 3i = (x + iy) - 2 - 3i = (x - 2) + i(y - 3) \]

### Step 2: Identify the real and imaginary parts
From the expression \( (x - 2) + i(y - 3) \), we identify:
- Real part: \( x - 2 \)
- Imaginary part: \( y - 3 \)

### Step 3: Use the definition of amplitude
The amplitude (or argument) of a complex number \( a + bi \) is given by:
\[ \tan(\theta) = \frac{b}{a} \]
In our case, we have:
\[ \tan\left(\frac{\pi}{4}\right) = \frac{y - 3}{x - 2} \]

### Step 4: Evaluate \( \tan\left(\frac{\pi}{4}\right) \)
We know that:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Thus, we can set up the equation:
\[ 1 = \frac{y - 3}{x - 2} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ y - 3 = x - 2 \]

### Step 6: Rearrange the equation
Rearranging the equation, we can express the relation between \( x \) and \( y \):
\[ y = x + 1 \]

### Final Relation
The relation between \( x \) and \( y \) is:
\[ y = x + 1 \]

---