If A and B represent the complex numbers `z_(1)` and `z_(2)` such that `|z_(1)+z_(2)|=|z_(1)-z_(2)|`, then the circumcenter of `triangleOAB`, where O is the origin, is

Options

A. `(z_(1)+z_(2))/(3)`
B. `(z_(1)+z_(2))/(2)`
C. `(z_(1)-z_(2))/(2)`
D. none of these

Explanation

To solve the problem, we need to analyze the given condition involving the complex numbers \( z_1 \) and \( z_2 \). The condition states that \( |z_1 + z_2| = |z_1 - z_2| \).

### Step-by-Step Solution:

1. **Understanding the Condition**:
We start with the condition \( |z_1 + z_2| = |z_1 - z_2| \). This implies that the distance from the origin to the point \( z_1 + z_2 \) is equal to the distance from the origin to the point \( z_1 - z_2 \).

2. **Squaring Both Sides**:
To simplify the equation, we square both sides:
\[ |z_1 + z_2|^2 = |z_1 - z_2|^2 \]
This leads to:
\[ (z_1 + z_2)(\overline{z_1 + z_2}) = (z_1 - z_2)(\overline{z_1 - z_2}) \]

3. **Expanding Both Sides**:
Expanding both sides, we have:
\[ |z_1|^2 + |z_2|^2 + z_1 \overline{z_2} + z_2 \overline{z_1} = |z_1|^2 + |z_2|^2 - z_1 \overline{z_2} - z_2 \overline{z_1} \]

4. **Rearranging the Equation**:
By rearranging the equation, we can eliminate \( |z_1|^2 + |z_2|^2 \) from both sides:
\[ z_1 \overline{z_2} + z_2 \overline{z_1} + z_1 \overline{z_2} + z_2 \overline{z_1} = 0 \]
This simplifies to:
\[ 2(z_1 \overline{z_2} + z_2 \overline{z_1}) = 0 \]
Thus, we have:
\[ z_1 \overline{z_2} + z_2 \overline{z_1} = 0 \]

5. **Conclusion About the Angles**:
The equation \( z_1 \overline{z_2} + z_2 \overline{z_1} = 0 \) implies that the angle \( \angle AOB \) formed by the vectors \( z_1 \) and \( z_2 \) is \( \frac{\pi}{2} \) (90 degrees). This means that the points \( A \) and \( B \) are at right angles with respect to the origin \( O \).

6. **Finding the Circumcenter**:
In a triangle where one angle is \( 90^\circ \), the circumcenter is located at the midpoint of the hypotenuse. The hypotenuse in this case is the line segment connecting points \( A \) and \( B \).

Therefore, the circumcenter \( C \) of triangle \( OAB \) is given by:
\[ C = \frac{z_1 + z_2}{2} \]

### Final Answer:
The circumcenter of triangle \( OAB \) is \( \frac{z_1 + z_2}{2} \).