The points representing the complex numbers z for which `|z+4|^(2)-|z-4|^(2)=8` lie on

Options

A. a straight line parallel to x-axis
B. a straight line parallel to y-axis
C. a circle with center as origin
D. a circle with center other than the origin.

Explanation

To solve the equation \( |z + 4|^2 - |z - 4|^2 = 8 \) and determine the geometric representation of the complex numbers \( z \), we can follow these steps:

### Step 1: Express \( z \) in terms of its real and imaginary parts
Let \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.

### Step 2: Write the expressions for the moduli
We need to compute \( |z + 4|^2 \) and \( |z - 4|^2 \):
- \( |z + 4|^2 = |(x + 4) + iy|^2 = (x + 4)^2 + y^2 \)
- \( |z - 4|^2 = |(x - 4) + iy|^2 = (x - 4)^2 + y^2 \)

### Step 3: Substitute the moduli into the equation
Substituting these into the given equation:
\[ |z + 4|^2 - |z - 4|^2 = (x + 4)^2 + y^2 - ((x - 4)^2 + y^2) = 8 \]

### Step 4: Simplify the equation
The \( y^2 \) terms cancel out:
\[ (x + 4)^2 - (x - 4)^2 = 8 \]
Now, expand both squares:
\[ (x^2 + 8x + 16) - (x^2 - 8x + 16) = 8 \]
This simplifies to:
\[ 8x + 8x = 8 \]
\[ 16x = 8 \]

### Step 5: Solve for \( x \)
Dividing both sides by 16:
\[ x = \frac{8}{16} = \frac{1}{2} \]

### Step 6: Interpret the result
Since \( x = \frac{1}{2} \) is a constant, the points representing the complex numbers \( z \) lie on the vertical line \( x = \frac{1}{2} \) in the complex plane.

### Final Result
The points representing the complex numbers \( z \) for which \( |z + 4|^2 - |z - 4|^2 = 8 \) lie on the vertical line \( x = \frac{1}{2} \).

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