If a, b, c are in H.P. then the value of `((1)/(b) + (1)/(c) - (1)/(a)) ((1)/(c) + (1)/(a) - (1)/(b))` is

Options

A. `(2)/(bc) - (1)/(b^(2))`
B. `(3)/(b^(2)) - (2)/(ab)`
C. `(1)/(4) ((3)/(c^(2)) + (2)/(ca) - (1)/(a^(2)))`
D. none

Answer

To solve the problem, we start with the fact that \( a, b, c \) are in Harmonic Progression (H.P.). This means that the reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.).

### Step-by-step Solution:

1. **Understanding H.P. and A.P. Relationship**:
Since \( a, b, c \) are in H.P., we can express the relationship of their reciprocals:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
This implies:
\[ \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b} \]

2. **Setting Up the Expression**:
We need to evaluate the expression:
\[ \left( \frac{1}{b} + \frac{1}{c} - \frac{1}{a} \right) \left( \frac{1}{c} + \frac{1}{a} - \frac{1}{b} \right) \]

3. **Substituting the Relationships**:
We can substitute the relationships derived from the H.P. condition into the expression:
- From the first part:
\[ \frac{1}{b} + \frac{1}{c} - \frac{1}{a} = \frac{1}{b} + \left( \frac{1}{b} + \frac{1}{a} \right) - \frac{1}{a} = 2 \cdot \frac{1}{b} \]
- For the second part:
\[ \frac{1}{c} + \frac{1}{a} - \frac{1}{b} = \left( \frac{1}{b} + \frac{1}{c} \right) - \frac{1}{b} = \frac{1}{c} \]

4. **Combining the Results**:
Now substituting these results back into the expression gives us:
\[ \left( 2 \cdot \frac{1}{b} \right) \left( \frac{1}{c} \right) = \frac{2}{bc} \]

5. **Final Expression**:
Thus, the value of the given expression is:
\[ \frac{2}{bc} \]

### Final Answer:
The value of the expression is \( \frac{2}{bc} \).