If `z_(1),z_(2)` are tow complex numberes `(z_(1) ne z_(2))` satisfying `|z_(1)^(2)- z_(2)^(2)|=|barz_(1)^(2)+barz_(2)^(2) - 2barz_(1)barz_(2)|`, then

Options

A. `(z_(1))/(z_(2))` is purely imaginary
B. `(z_(1))/(z_(2))` is purely real
C. `|argz_(1) - argz_(2)| = pi`
D. `|arg z_(1)-argz_(2)|=(pi)/(2)`

Explanation

To solve the problem step by step, we start with the given equation involving two complex numbers \( z_1 \) and \( z_2 \):

### Step 1: Write the given equation
We have:
\[ |z_1^2 - z_2^2| = |\overline{z_1^2} + \overline{z_2^2} - 2\overline{z_1}\overline{z_2}| \]

### Step 2: Simplify the left-hand side
The left-hand side can be factored using the difference of squares:
\[ |z_1^2 - z_2^2| = |(z_1 - z_2)(z_1 + z_2| \]

### Step 3: Simplify the right-hand side
The right-hand side can be rewritten using properties of conjugates:
\[ \overline{z_1^2} = \overline{z_1}^2 \quad \text{and} \quad \overline{z_2^2} = \overline{z_2}^2 \]
Thus, we can write:
\[ |\overline{z_1}^2 + \overline{z_2}^2 - 2\overline{z_1}\overline{z_2}| = |(\overline{z_1} - \overline{z_2})^2| \]

### Step 4: Equate both sides
Now we have:
\[ |(z_1 - z_2)(z_1 + z_2)| = |(\overline{z_1} - \overline{z_2})^2| \]

### Step 5: Use properties of modulus
The modulus of a product is the product of the moduli:
\[ |z_1 - z_2| |z_1 + z_2| = | \overline{z_1} - \overline{z_2}|^2 \]
Since \( | \overline{z_1} - \overline{z_2}| = |z_1 - z_2| \), we can write:
\[ |z_1 - z_2| |z_1 + z_2| = |z_1 - z_2|^2 \]

### Step 6: Divide both sides by \( |z_1 - z_2| \)
Assuming \( z_1 \neq z_2 \), we can divide both sides by \( |z_1 - z_2| \):
\[ |z_1 + z_2| = |z_1 - z_2| \]

### Step 7: Interpret the result geometrically
The equation \( |z_1 + z_2| = |z_1 - z_2| \) implies that the points \( z_1 \) and \( z_2 \) are equidistant from the origin. This means that the line segment connecting \( z_1 \) and \( z_2 \) is perpendicular to the line connecting the origin to the midpoint of \( z_1 \) and \( z_2 \).

### Step 8: Conclusion
From the above steps, we conclude that:
- The complex numbers \( z_1 \) and \( z_2 \) are such that \( \frac{z_1}{z_2} \) is purely imaginary.
- The argument of \( z_1 - z_2 \) and \( z_2 - z_1 \) differ by \( \frac{\pi}{2} \).