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

Options

A. `x+y-1=0`
B. `x-y-1=0`
C. `x+y+1=0`
D. `x-y+1=0`

Answer

To solve the problem, we need to find the locus of the complex number \( z = x + iy \) given that the amplitude (or argument) of \( z - 2 - 3i \) is \( \frac{\pi}{4} \).

### Step-by-Step Solution:

1. **Express the complex number**:
We start with the complex number \( z - 2 - 3i \). Let \( z = x + iy \). Thus, we have:
\[ z - 2 - 3i = (x - 2) + i(y - 3) \]

2. **Set the argument**:
The argument (or amplitude) of a complex number \( a + bi \) is given by \( \tan^{-1}(\frac{b}{a}) \). Therefore, we need to set up the equation:
\[ \tan^{-1}\left(\frac{y - 3}{x - 2}\right) = \frac{\pi}{4} \]

3. **Use the tangent function**:
Since \( \tan\left(\frac{\pi}{4}\right) = 1 \), we can equate:
\[ \frac{y - 3}{x - 2} = 1 \]

4. **Cross-multiply to eliminate the fraction**:
Cross-multiplying gives us:
\[ y - 3 = x - 2 \]

5. **Rearrange the equation**:
Rearranging the equation results in:
\[ x - y + 1 = 0 \]

6. **Identify the locus**:
The equation \( x - y + 1 = 0 \) represents a straight line in the Cartesian plane.

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