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

Options

A. a. `3/(b^2) - (2)/(ab)`
B. b. `a^2 + b^2 + c^2`
C. c. `(4)/(ac) - 3/(b^2)`
D. d. both (a) and (c)

Answer

To solve the problem, we need to find the value of the expression:

\[ \left( \frac{1}{b} + \frac{1}{a} - \frac{1}{c} \right) \left( \frac{1}{b} + \frac{1}{c} - \frac{1}{a} \right) \]

Given that \(a\), \(b\), and \(c\) are in Harmonic Progression (H.P.), we know that:

\[ \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \text{ are in Arithmetic Progression (A.P.)} \]

This means that:

\[ \frac{1}{b} = \frac{1}{2} \left( \frac{1}{a} + \frac{1}{c} \right) \]

### Step 1: Rewrite the expressions

We can rewrite the expressions in the brackets using the property of A.P.:

1. The first expression becomes:
\[ \frac{1}{b} + \frac{1}{a} - \frac{1}{c} = \frac{1}{b} + \frac{1}{a} - \left(2 \cdot \frac{1}{b} - \frac{1}{a}\right) = \frac{1}{b} + \frac{1}{a} - 2 \cdot \frac{1}{b} + \frac{1}{a} = 2 \cdot \frac{1}{a} - \frac{1}{b} \]

2. The second expression becomes:
\[ \frac{1}{b} + \frac{1}{c} - \frac{1}{a} = \frac{1}{b} + \left(2 \cdot \frac{1}{b} - \frac{1}{a}\right) - \frac{1}{a} = 2 \cdot \frac{1}{b} - \frac{1}{a} \]

### Step 2: Multiply the two expressions

Now we multiply the two results:

\[ \left(2 \cdot \frac{1}{a} - \frac{1}{b}\right) \left(2 \cdot \frac{1}{b} - \frac{1}{a}\right) \]

### Step 3: Expand the product

Using the distributive property (FOIL method):

\[ = 4 \cdot \frac{1}{ab} - 2 \cdot \frac{1}{a^2} - 2 \cdot \frac{1}{b^2} + \frac{1}{ab} \]

### Step 4: Combine like terms

Combining the terms gives:

\[ = \frac{5}{ab} - 2 \left(\frac{1}{a^2} + \frac{1}{b^2}\right) \]

### Step 5: Use the relationship of H.P.

Since \(a\), \(b\), and \(c\) are in H.P., we can use the relationship:

\[ \frac{1}{a} + \frac{1}{c} = \frac{2}{b} \]

This implies:

\[ \frac{1}{a^2} + \frac{1}{c^2} = \frac{2}{b^2} \]

### Final Result

Thus, the final expression simplifies to:

\[ = \frac{3}{b^2} \]

### Conclusion

The value of the expression is:

\[ \frac{3}{b^2} \]