if `Im((z+2i)/(z+2))= 0` then z lies on the curve :

Options

A. `x^(2) + y^(2) + 2 x + 2y = 0`
B. `x^(2) + y^(2) - 2 x = 0`
C. `x + y + 2 =0`
D. None of these

Answer

To solve the problem where we need to find the curve on which the complex number \( z \) lies, given that \( \text{Im}\left(\frac{z + 2i}{z + 2}\right) = 0 \), we can follow these steps:

### Step 1: Substitute \( z \)
Assume \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Rewrite the expression
Now, substitute \( z \) into the expression:
\[ \frac{z + 2i}{z + 2} = \frac{(x + iy) + 2i}{(x + iy) + 2} = \frac{x + i(y + 2)}{(x + 2) + iy} \]

### Step 3: Rationalize the denominator
To find the imaginary part, we need to rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{x + i(y + 2)}{(x + 2) + iy} \cdot \frac{(x + 2) - iy}{(x + 2) - iy} \]

### Step 4: Multiply out the numerator
The numerator becomes:
\[ (x + i(y + 2))((x + 2) - iy) = x(x + 2) + 2i + y^2 + 2y - ixy - 2y \]
This simplifies to:
\[ x(x + 2) + 2y + 2i - xy \]

### Step 5: Multiply out the denominator
The denominator becomes:
\[ (x + 2)^2 + y^2 \]

### Step 6: Write the full expression
Now we can write the full expression:
\[ \frac{x(x + 2) + 2y + 2i - xy}{(x + 2)^2 + y^2} \]

### Step 7: Identify the imaginary part
The imaginary part of the expression is:
\[ \text{Im}\left(\frac{x(x + 2) + 2y - xy + 2i}{(x + 2)^2 + y^2}\right) = \frac{2 + 2y - xy}{(x + 2)^2 + y^2} \]

### Step 8: Set the imaginary part to zero
Since we are given that the imaginary part is equal to zero:
\[ 2 + 2y - xy = 0 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ xy - 2y - 2 = 0 \]

### Step 10: Factor the equation
We can factor this equation:
\[ y(x - 2) = 2 \]

### Conclusion
Thus, the equation \( y(x - 2) = 2 \) describes the curve on which \( z \) lies.

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