If `"Re"((z-8i)/(z+6))=0`, then z 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 analyze the equation given:

**Given**: \( \text{Re}\left(\frac{z - 8i}{z + 6}\right) = 0 \)

**Step 1: Express \( z \) in terms of \( x \) and \( y \)**
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

**Step 2: Substitute \( z \) into the expression**
Now, we substitute \( z \) into the expression:
\[ \frac{z - 8i}{z + 6} = \frac{(x + iy) - 8i}{(x + iy) + 6} = \frac{x + i(y - 8)}{(x + 6) + iy} \]

**Step 3: Separate real and imaginary parts**
This can be rewritten as:
\[ \frac{x + i(y - 8)}{(x + 6) + iy} \]

**Step 4: Rationalize the denominator**
To eliminate the imaginary part from the denominator, we multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(x + i(y - 8))((x + 6) - iy)}{((x + 6) + iy)((x + 6) - iy)} \]

**Step 5: Expand the numerator**
Expanding the numerator:
\[ = (x(x + 6) - xy) + i((y - 8)(x + 6) - yx) \]
This simplifies to:
\[ = (x^2 + 6x - xy) + i((y - 8)(x + 6) - yx) \]

**Step 6: Expand the denominator**
The denominator simplifies to:
\[ (x + 6)^2 + y^2 \]

**Step 7: Write the expression**
Thus, we have:
\[ \frac{(x^2 + 6x - xy) + i((y - 8)(x + 6) - yx)}{(x + 6)^2 + y^2} \]

**Step 8: Set the real part to zero**
Since we need the real part to be zero:
\[ x^2 + 6x - xy = 0 \]

**Step 9: Rearrange the equation**
Rearranging gives:
\[ x^2 + 6x = xy \]

**Step 10: Further simplify**
Rearranging further, we get:
\[ x^2 + 6x - xy = 0 \implies x^2 + 6x - 8y = 0 \]

**Step 11: Identify the curve**
This is a quadratic equation in \( x \). The equation represents a parabola in the \( xy \)-plane.

**Final Result**: The complex number \( z \) lies on the curve defined by the equation:
\[ x^2 + 6x - 8y = 0 \]