If ` z = x + iy, x , y in R ` , then the louts `Im (( z - 2 ) /(z + i)) = (1 ) /(2) ` represents : ( where ` i= sqrt ( - 1)) `

Options

A. Straight line
B. Circle
C. Parabola
D. Hyperbola

Explanation

To solve the problem, we need to analyze the expression given and find the imaginary part of the complex expression. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Define the complex number**:
Let \( z = x + iy \), where \( x, y \in \mathbb{R} \).

2. **Set up the equation**:
We need to find the imaginary part of the expression:
\[ \text{Im} \left( \frac{z - 2}{z + i} \right) = \frac{1}{2} \]

3. **Substitute \( z \)**:
Substitute \( z \) into the expression:
\[ \frac{z - 2}{z + i} = \frac{(x + iy) - 2}{(x + iy) + i} = \frac{(x - 2) + iy}{x + (y + 1)i} \]

4. **Rationalize the denominator**:
Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{((x - 2) + iy)(x - (y + 1)i)}{(x + (y + 1)i)(x - (y + 1)i)} \]

5. **Calculate the denominator**:
The denominator simplifies to:
\[ x^2 + (y + 1)^2 \]

6. **Calculate the numerator**:
The numerator expands as follows:
\[ (x - 2)x - (x - 2)(y + 1)i + iyx - iy(y + 1)i \]
Simplifying this gives:
\[ (x^2 - 2x + y^2 + y + 1) + i(yx - (x - 2)(y + 1)) \]

7. **Extract the imaginary part**:
The imaginary part of the expression is:
\[ \frac{yx - (x - 2)(y + 1)}{x^2 + (y + 1)^2} \]

8. **Set the imaginary part equal to \( \frac{1}{2} \)**:
\[ \frac{yx - (x - 2)(y + 1)}{x^2 + (y + 1)^2} = \frac{1}{2} \]

9. **Cross multiply**:
\[ 2(yx - (x - 2)(y + 1)) = x^2 + (y + 1)^2 \]

10. **Expand and simplify**:
Expanding both sides leads to:
\[ 2yx - 2xy - 2y + 4 = x^2 + y^2 + 2y + 1 \]
Rearranging gives:
\[ x^2 + y^2 - 2y + 2x - 3 = 0 \]

11. **Identify the conic section**:
The equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \) represents a circle. Here, we can see that it is indeed a circle.

### Conclusion:
The equation represents a circle.