Im `((2z+1)/(iz+1))=5` represents

Options

A. a circle
B. a straight line
C. a parabola
D. an ellipse

Answer

To solve the problem, we need to analyze the equation given by the imaginary part of the complex expression \(\frac{2z + 1}{iz + 1} = 5\). We will substitute \(z\) with \(x + iy\) (where \(x\) is the real part and \(y\) is the imaginary part) and then find the locus of \(z\).

### Step-by-Step Solution:

1. **Substitute \(z\)**:
Let \(z = x + iy\). Then, we have:
\[ 2z + 1 = 2(x + iy) + 1 = 2x + 1 + 2iy \]
\[ iz + 1 = i(x + iy) + 1 = -y + ix + 1 = 1 - y + ix \]

2. **Set up the equation**:
The equation becomes:
\[ \frac{2x + 1 + 2iy}{1 - y + ix} = 5 \]

3. **Multiply both sides by the denominator**:
\[ 2x + 1 + 2iy = 5(1 - y + ix) \]
Expanding the right side:
\[ 2x + 1 + 2iy = 5 - 5y + 5ix \]

4. **Separate real and imaginary parts**:
From the equation, we can separate the real and imaginary parts:
- Real part: \(2x + 1 = 5 - 5y\)
- Imaginary part: \(2y = 5x\)

5. **Rearranging the equations**:
From the real part:
\[ 2x + 5y = 4 \quad \text{(1)} \]
From the imaginary part:
\[ y = \frac{5}{2}x \quad \text{(2)} \]

6. **Substituting equation (2) into equation (1)**:
Substitute \(y\) from (2) into (1):
\[ 2x + 5\left(\frac{5}{2}x\right) = 4 \]
Simplifying:
\[ 2x + \frac{25}{2}x = 4 \]
\[ \frac{4x + 25x}{2} = 4 \]
\[ \frac{29x}{2} = 4 \implies 29x = 8 \implies x = \frac{8}{29} \]

7. **Finding \(y\)**:
Substitute \(x\) back into equation (2):
\[ y = \frac{5}{2}\left(\frac{8}{29}\right) = \frac{40}{29} \]

8. **Conclusion**:
The locus of \(z\) is represented by the equations \(2x + 5y = 4\) and \(y = \frac{5}{2}x\), which describe a straight line.

### Final Answer:
The locus of \(z\) represents a straight line.