If `P and Q` are represented by the complex numbers `z_1 and z_2` such that `|(1)/(z_2) + (1)/(z_1)|= |(1)/(z_2) - (1)/(z_1)|`, then the circumstance of `Delta OPQ` (where O is the origin) is

Options

A. `(1)/(2) (z_1 - z_2)`
B. `(1)/(3) (z_1 + z_2)`
C. `(1)/(2) (z_1 + z_2)`
D. `(1)/(3) (z_1 - z_2)`

Answer

To solve the problem, we start with the given condition:

\[ \left| \frac{1}{z_2} + \frac{1}{z_1} \right| = \left| \frac{1}{z_2} - \frac{1}{z_1} \right| \]

### Step 1: Rewrite the equation
We can rewrite the left-hand side and right-hand side using a common denominator:

\[ \left| \frac{z_1 + z_2}{z_1 z_2} \right| = \left| \frac{z_1 - z_2}{z_1 z_2} \right| \]

### Step 2: Remove the denominator
Since \( z_1 z_2 \) is non-zero (as \( z_1 \) and \( z_2 \) are complex numbers representing points P and Q), we can multiply both sides by \( |z_1 z_2| \):

\[ |z_1 + z_2| = |z_1 - z_2| \]

### Step 3: Use the property of moduli
The equation \( |z_1 + z_2| = |z_1 - z_2| \) indicates that the points represented by \( z_1 \) and \( z_2 \) are equidistant from the origin. This implies that the points form a right triangle with the origin.

### Step 4: Interpret geometrically
The condition \( |z_1 + z_2| = |z_1 - z_2| \) geometrically means that the angle between the vectors \( z_1 \) and \( z_2 \) is \( 90^\circ \). Therefore, the triangle formed by the points O (origin), P (point represented by \( z_1 \)), and Q (point represented by \( z_2 \)) is a right triangle.

### Step 5: Find the circumcenter
In a right triangle, the circumcenter (the center of the circumcircle) is located at the midpoint of the hypotenuse. The hypotenuse in this case is the line segment joining points P and Q.

The circumcenter \( C \) can be calculated as:

\[ C = \frac{z_1 + z_2}{2} \]

### Conclusion
Thus, the circumcenter of triangle OPQ is given by:

\[ C = \frac{z_1 + z_2}{2} \]