For two non - zero complex numbers A and B, if `A+(1)/(B)=barA and (1)/(A)+B=barB`, then

Options

A. A is purely real
B. B is purely real
C. `|AB|=(1)/(2)`
D. `|A|=|B|`

Answer

To solve the problem involving two non-zero complex numbers \( A \) and \( B \) given the equations \( A + \frac{1}{B} = \bar{A} \) and \( \frac{1}{A} + B = \bar{B} \), we can follow these steps:

### Step 1: Rewrite the equations
We start with the equations:
1. \( A + \frac{1}{B} = \bar{A} \)
2. \( \frac{1}{A} + B = \bar{B} \)

### Step 2: Isolate the terms
From the first equation, we can isolate \( \frac{1}{B} \):
\[ \frac{1}{B} = \bar{A} - A \]
From the second equation, we can isolate \( B \):
\[ B = \bar{B} - \frac{1}{A} \]

### Step 3: Analyze the imaginary parts
We know that \( A - \bar{A} \) is purely imaginary. Therefore, \( \bar{A} - A \) is also purely imaginary. This implies that \( \frac{1}{B} \) must be purely imaginary, which means \( B \) must also be purely imaginary.

Similarly, from the second equation, \( B - \bar{B} \) is purely imaginary, leading us to conclude that \( A \) is also purely imaginary.

### Step 4: Let \( A \) and \( B \) be purely imaginary
We can express \( A \) and \( B \) as:
\[ A = i a \quad \text{and} \quad B = i b \]
where \( a \) and \( b \) are real numbers.

### Step 5: Substitute into the equations
Substituting \( A \) and \( B \) into the first equation:
\[ i a + \frac{1}{i b} = -i a \]
This simplifies to:
\[ i a - \frac{i}{b} = -i a \]
Rearranging gives:
\[ 2i a = \frac{i}{b} \]
Thus:
\[ 2 a b = 1 \quad \text{(1)} \]

Now substituting into the second equation:
\[ \frac{1}{i a} + i b = -i b \]
This simplifies to:
\[ -\frac{i}{a} + i b = -i b \]
Rearranging gives:
\[ -\frac{i}{a} + 2i b = 0 \]
Thus:
\[ \frac{1}{a} = 2b \quad \text{(2)} \]

### Step 6: Solve the equations
From equation (1):
\[ ab = \frac{1}{2} \]
From equation (2):
\[ a = \frac{1}{2b} \]
Substituting \( a \) from (2) into (1):
\[ \left(\frac{1}{2b}\right)b = \frac{1}{2} \]
This simplifies to:
\[ \frac{1}{2} = \frac{1}{2} \]
This is consistent, and we can find \( ab \):
\[ ab = \frac{1}{2} \]

### Step 7: Find \( |AB| \)
Since \( A = i a \) and \( B = i b \):
\[ |AB| = |i a| |i b| = |a| |b| = ab \]
Thus:
\[ |AB| = \frac{1}{2} \]

### Conclusion
The final answer is:
\[ |AB| = \frac{1}{2} \]