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 problem, we need to analyze the given equation involving complex numbers. The equation is:

\[ |z + 4|^2 - |z - 4|^2 = 8 \]

### Step 1: Substitute \( z \)
Let \( z = a + ib \), where \( a \) is the real part and \( b \) is the imaginary part of the complex number \( z \).

### Step 2: Write the Modulus
We can express the moduli as follows:
\[ |z + 4|^2 = |(a + 4) + ib|^2 = (a + 4)^2 + b^2 \]
\[ |z - 4|^2 = |(a - 4) + ib|^2 = (a - 4)^2 + b^2 \]

### Step 3: Substitute into the Equation
Now, substitute these expressions into the original equation:
\[ (a + 4)^2 + b^2 - \left((a - 4)^2 + b^2\right) = 8 \]

### Step 4: Simplify the Equation
The \( b^2 \) terms cancel out:
\[ (a + 4)^2 - (a - 4)^2 = 8 \]

### Step 5: Expand the Squares
Now expand both squares:
\[ (a^2 + 8a + 16) - (a^2 - 8a + 16) = 8 \]

### Step 6: Combine Like Terms
This simplifies to:
\[ 8a + 8a = 8 \]
\[ 16a = 8 \]

### Step 7: Solve for \( a \)
Now, divide both sides by 16:
\[ a = \frac{8}{16} = \frac{1}{2} \]

### Step 8: Determine \( b \)
Since there are no restrictions on \( b \) from the original equation, \( b \) can take any value. Therefore, we can express \( z \) as:
\[ z = \frac{1}{2} + ib \]

### Conclusion
The points representing the complex numbers \( z \) lie on a vertical line at \( x = \frac{1}{2} \) in the complex plane, which is parallel to the y-axis.

### Final Answer
The points representing the complex numbers \( z \) lie on a straight line parallel to the y-axis and passing through the point \( \left(\frac{1}{2}, 0\right) \).

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