Which one of the following is correct ? If z and w are complex numbers and `bar(w)` denotes the conjugate of w, then `|z+w| = |z-w|` holds only if `:`

Options

A. z=0 or w=0
B. z=0
C. `z. bar(w)` is purely real
D. `z. bar(w)` is purely imaginary

Answer

To solve the problem, we need to find the condition under which the modulus of the sum of two complex numbers \( z \) and \( w \) is equal to the modulus of their difference.

Let’s denote the complex numbers as:
- \( z = a + ib \)
- \( w = c + id \)

We need to determine when:
\[ |z + w| = |z - w| \]

### Step 1: Express \( z + w \) and \( z - w \)
First, we calculate \( z + w \) and \( z - w \):
\[ z + w = (a + ib) + (c + id) = (a + c) + i(b + d) \]
\[ z - w = (a + ib) - (c + id) = (a - c) + i(b - d) \]

### Step 2: Calculate the moduli
Next, we calculate the moduli of both expressions:
\[ |z + w| = \sqrt{(a + c)^2 + (b + d)^2} \]
\[ |z - w| = \sqrt{(a - c)^2 + (b - d)^2} \]

### Step 3: Set the moduli equal
Now, we set the two moduli equal:
\[ \sqrt{(a + c)^2 + (b + d)^2} = \sqrt{(a - c)^2 + (b - d)^2} \]

### Step 4: Square both sides
To eliminate the square roots, we square both sides:
\[ (a + c)^2 + (b + d)^2 = (a - c)^2 + (b - d)^2 \]

### Step 5: Expand both sides
Expanding both sides gives:
\[ (a^2 + 2ac + c^2 + b^2 + 2bd + d^2) = (a^2 - 2ac + c^2 + b^2 - 2bd + d^2) \]

### Step 6: Simplify the equation
Now, we can simplify the equation by canceling out common terms:
\[ 2ac + 2bd = -2ac - 2bd \]

### Step 7: Combine like terms
Combining like terms results in:
\[ 4ac + 4bd = 0 \]

### Step 8: Factor out common terms
Factoring out the common terms gives:
\[ ac + bd = 0 \]

### Conclusion
Thus, the condition under which \( |z + w| = |z - w| \) holds is:
\[ ac + bd = 0 \]
This means that the real part of the product of \( z \) and the conjugate of \( w \) is zero, indicating that the product is purely imaginary.

### Final Answer
The correct condition is that \( z \cdot \overline{w} \) is purely imaginary.