If the amplitude of `z-2-3i " is " (pi)/(4)`, then find the locus of `z= x+ yi`

Answer

To find the locus of the complex number \( z = x + yi \) given that the amplitude of \( z - 2 - 3i \) is \( \frac{\pi}{4} \), we can follow these steps:

### Step-by-step Solution:

1. **Define the Complex Number**:
Let \( z = x + yi \).

2. **Express \( z - 2 - 3i \)**:
We can rewrite \( z - 2 - 3i \) as:
\[ z - 2 - 3i = (x + yi) - (2 + 3i) = (x - 2) + (y - 3)i \]

3. **Identify the Amplitude**:
The amplitude (or argument) of a complex number \( a + bi \) is given by:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
In our case, we have:
\[ a = x - 2 \quad \text{and} \quad b = y - 3 \]
Thus, the amplitude \( \theta \) can be expressed as:
\[ \theta = \tan^{-1}\left(\frac{y - 3}{x - 2}\right) \]

4. **Set the Amplitude Equal to \( \frac{\pi}{4} \)**:
Given that the amplitude is \( \frac{\pi}{4} \), we have:
\[ \tan\left(\frac{\pi}{4}\right) = 1 \]
Therefore, we can set up the equation:
\[ \frac{y - 3}{x - 2} = 1 \]

5. **Solve for the Relationship Between \( x \) and \( y \)**:
From the equation \( \frac{y - 3}{x - 2} = 1 \), we can cross-multiply to obtain:
\[ y - 3 = x - 2 \]
Rearranging gives:
\[ y = x + 1 \]

6. **Write the Locus Equation**:
The equation \( y = x + 1 \) represents a straight line. We can also express it in standard form:
\[ x - y + 1 = 0 \]

### Final Answer:
The locus of the complex number \( z \) is the straight line given by the equation:
\[ x - y + 1 = 0 \]