The points representing the complex numbers z for which `| z + 3i|^(2) - | z - 3| ^(2) = 6 ` lie on the line given by

Options

A. x + y = 1
B. x + y = - 1
C. x + y = 2
D. x + y = 0

Answer

To solve the equation \( |z + 3i|^2 - |z - 3|^2 = 6 \) where \( z \) is a complex number, we can follow these steps:

### Step 1: Represent the complex number
Let \( z = x + iy \), where \( x \) and \( y \) are real numbers.

### Step 2: Substitute \( z \) into the equation
We substitute \( z \) into the equation:
\[ |z + 3i|^2 - |z - 3|^2 = 6 \]
This becomes:
\[ |x + iy + 3i|^2 - |x - 3 + iy|^2 = 6 \]

### Step 3: Simplify the moduli
Now we simplify each modulus:
1. \( |z + 3i| = |x + i(y + 3)| = \sqrt{x^2 + (y + 3)^2} \)
2. \( |z - 3| = |(x - 3) + iy| = \sqrt{(x - 3)^2 + y^2} \)

Now squaring both sides:
\[ |z + 3i|^2 = x^2 + (y + 3)^2 = x^2 + y^2 + 6y + 9 \]
\[ |z - 3|^2 = (x - 3)^2 + y^2 = x^2 - 6x + 9 + y^2 \]

### Step 4: Substitute back into the equation
Substituting these into the original equation gives:
\[ (x^2 + y^2 + 6y + 9) - (x^2 - 6x + 9 + y^2) = 6 \]

### Step 5: Simplify the equation
Now simplify:
\[ x^2 + y^2 + 6y + 9 - x^2 + 6x - 9 - y^2 = 6 \]
This simplifies to:
\[ 6y + 6x = 6 \]

### Step 6: Divide through by 6
Dividing the entire equation by 6 gives:
\[ y + x = 1 \]

### Step 7: Rearranging the equation
This can be rearranged to:
\[ x + y = 1 \]

### Conclusion
The points representing the complex numbers \( z \) lie on the line given by the equation:
\[ x + y = 1 \]