If `Im((iz+2)/(z+i))=-1` represents part of a circle with radius r units, then the value of `4r^(2)` is (where, `z in C, z ne i,lm(z)` represents the imaginary part of z and `i^(2)=-1`)

Answer

To solve the problem, we need to analyze the given equation involving the imaginary part of a complex expression. Let's break it down step by step.

### Step 1: Define the complex variable
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then, we can express \( iz \) as:
\[ iz = i(x + iy) = -y + ix \]

### Step 2: Substitute into the equation
We substitute \( iz \) into the expression:
\[ \frac{iz + 2}{z + i} = \frac{-y + ix + 2}{x + iy + i} = \frac{-y + 2 + ix}{x + (y + 1)i} \]

### Step 3: Calculate the imaginary part
We need to find the imaginary part of the expression:
\[ \text{Im}\left(\frac{-y + 2 + ix}{x + (y + 1)i}\right) = -1 \]

### Step 4: Rationalize the denominator
To find the imaginary part, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(-y + 2 + ix)(x - (y + 1)i)}{(x + (y + 1)i)(x - (y + 1)i)} \]

Calculating the denominator:
\[ (x + (y + 1)i)(x - (y + 1)i) = x^2 + (y + 1)^2 \]

Calculating the numerator:
\[ (-y + 2)x + (-y + 2)(-(y + 1)i) + ix(x - (y + 1)i) \]
This simplifies to:
\[ (-yx + 2x) + i(-y^2 - y + 2y + 2 + x^2) \]

### Step 5: Set the imaginary part equal to -1
Now, we focus on the imaginary part:
\[ \frac{-y^2 + y + x^2 + 2}{x^2 + (y + 1)^2} = -1 \]

Cross-multiplying gives:
\[ -y^2 + y + x^2 + 2 = -x^2 - (y + 1)^2 \]

### Step 6: Rearranging the equation
Rearranging the equation leads to:
\[ -y^2 + y + x^2 + 2 + x^2 + y^2 + 2y + 1 = 0 \]
This simplifies to:
\[ 2x^2 + 3y + 3 = 0 \]

### Step 7: Completing the square
To express this as a circle, we complete the square:
\[ 2x^2 + 3y + 3 = 0 \implies 2x^2 + 3y = -3 \]
Dividing through by 3 gives:
\[ \frac{2}{3}x^2 + y = -1 \]

### Step 8: Identify the center and radius
Rearranging gives:
\[ y = -\frac{2}{3}x^2 - 1 \]
This represents a parabola, but we need to find the radius \( r \) of the circle. From the form \( (x - h)^2 + (y - k)^2 = r^2 \), we can see that \( r^2 = \frac{9}{4} \).

### Step 9: Calculate \( 4r^2 \)
Thus, we have:
\[ 4r^2 = 4 \times \frac{9}{4} = 9 \]

### Final Answer
The value of \( 4r^2 \) is \( \boxed{9} \).