If `"Re"((z-8i)/(z+6))=0`, then lies on the curve

Options

A. `x^(2)+y^(2)+6x-8y=0`
B. `4x-3y+24=0`
C. `x^(2)+y^(2)-8=0`
D. none of these

Answer

To solve the problem, we need to find the condition under which the real part of the expression \(\frac{z - 8i}{z + 6}\) equals zero. Here, \(z\) is a complex number which can be expressed as \(z = x + iy\), where \(x\) and \(y\) are real numbers.

### Step-by-Step Solution:

1. **Substitute \(z\) with \(x + iy\)**:
\[ z - 8i = (x + iy) - 8i = x + i(y - 8) \]
\[ z + 6 = (x + iy) + 6 = (x + 6) + iy \]

2. **Write the expression**:
\[ \frac{z - 8i}{z + 6} = \frac{x + i(y - 8)}{(x + 6) + iy} \]

3. **Multiply numerator and denominator by the conjugate of the denominator**:
\[ \frac{(x + i(y - 8))((x + 6) - iy)}{((x + 6) + iy)((x + 6) - iy)} \]

4. **Calculate the denominator**:
\[ ((x + 6) + iy)((x + 6) - iy) = (x + 6)^2 + y^2 \]

5. **Calculate the numerator**:
\[ (x + i(y - 8))((x + 6) - iy) = x(x + 6) - ixy + i(y - 8)(x + 6) + (y - 8)(-iy) \]
\[ = x(x + 6) + (y - 8)(-i^2) + i[(y - 8)(x + 6) - xy] \]
\[ = x(x + 6) + (y - 8) + i[(y - 8)(x + 6) - xy] \]

6. **Combine terms**:
\[ = (x^2 + 6x + y - 8) + i[(y - 8)(x + 6) - xy] \]

7. **Set the real part equal to zero**:
\[ x^2 + 6x + y - 8 = 0 \]

8. **Rearranging gives the equation of the curve**:
\[ x^2 + 6x + y - 8 = 0 \implies x^2 + 6x + y = 8 \]

9. **Complete the square for the \(x\) terms**:
\[ (x + 3)^2 - 9 + y = 8 \implies (x + 3)^2 + y = 17 \]

### Conclusion:
The equation \((x + 3)^2 + y = 17\) represents a parabola that opens upwards.