The equation `I m((i z-2)/(z-i))+1=0, z & epsi; C , z!=i` represents a part of a circle having radius equal to

Answer

To solve the problem, we need to analyze the given equation:

\[ \text{Im}\left(\frac{i z - 2}{z - i}\right) + 1 = 0 \]

where \( z \) is a complex number and \( z \neq i \). We want to find the radius of the circle represented by this equation.

### Step 1: Rewrite the equation

We can rewrite the equation as:

\[ \text{Im}\left(\frac{i z - 2}{z - i}\right) = -1 \]

### Step 2: Substitute \( z \)

Let \( z = x + iy \), where \( x \) and \( y \) are real numbers. Then:

\[ i z = i(x + iy) = -y + ix \]

So, we have:

\[ i z - 2 = (-y - 2) + ix \]

And for the denominator:

\[ z - i = (x + iy) - i = x + i(y - 1) \]

### Step 3: Substitute into the equation

Now substituting these into our equation:

\[ \frac{(-y - 2) + ix}{x + i(y - 1)} \]

### Step 4: Multiply numerator and denominator by the conjugate of the denominator

To simplify, we multiply the numerator and denominator by the conjugate of the denominator:

\[ \frac{((-y - 2) + ix)(x - i(y - 1))}{(x + i(y - 1))(x - i(y - 1))} \]

Calculating the denominator:

\[ (x + i(y - 1))(x - i(y - 1)) = x^2 + (y - 1)^2 \]

### Step 5: Expand the numerator

Now, expanding the numerator:

\[ ((-y - 2)x + (y - 1)i(-y - 2) + ix \cdot x - ix \cdot i(y - 1)) \]

This simplifies to:

\[ (-yx - 2x) + i(x^2 + y - 1 - y - 2) \]

### Step 6: Extract the imaginary part

The imaginary part of the fraction is:

\[ \text{Im}\left(\frac{(-y - 2)x + i(x^2 + y - 1)}{x^2 + (y - 1)^2}\right) = \frac{x^2 + y - 1}{x^2 + (y - 1)^2} \]

Setting this equal to \(-1\):

\[ \frac{x^2 + y - 1}{x^2 + (y - 1)^2} = -1 \]

### Step 7: Cross-multiply and simplify

Cross-multiplying gives:

\[ x^2 + y - 1 = - (x^2 + (y - 1)^2) \]

Expanding and simplifying leads to:

\[ 2x^2 + 2y^2 - 2y - 1 = 0 \]

### Step 8: Rearranging to the standard form of a circle

Rearranging gives:

\[ 2x^2 + 2y^2 - 2y = 1 \]

Dividing through by 2:

\[ x^2 + y^2 - y = \frac{1}{2} \]

Completing the square for \(y\):

\[ x^2 + (y - \frac{1}{2})^2 = \frac{1}{2} + \frac{1}{4} = \frac{3}{4} \]

### Step 9: Identify the center and radius

This represents a circle with center at \((0, \frac{1}{2})\) and radius:

\[ r = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \]

### Conclusion

Thus, the radius of the circle represented by the equation is:

\[ \frac{\sqrt{3}}{2} \]