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

Options

A. (a) `4/(ac) - 3/b^2`
B. (b) `4/(ac) + 3/b^2`
C. (c) `2/(ac) - 3/b^2`
D. (d) `2/(ac) - 1/b^2`

Answer

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

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

given that \(a\), \(b\), and \(c\) are in Harmonic Progression (H.P.).

### Step 1: Understanding Harmonic Progression
Since \(a\), \(b\), and \(c\) are in H.P., we know that:

\[ \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \]

are in Arithmetic Progression (A.P.). This means:

\[ \frac{1}{b} - \frac{1}{a} = \frac{1}{c} - \frac{1}{b} \]

Rearranging this gives us:

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

Let’s denote this as Equation (1).

### Step 2: Substitute into the Expression
Now, we substitute the value from Equation (1) into the expression we need to evaluate:

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

### Step 3: Simplifying the First Term
For the first term:

\[ \frac{1}{b} + \frac{1}{c} - \frac{1}{a} = \frac{1}{b} + \left( \frac{2}{b} - \frac{1}{a} \right) - \frac{1}{a} = \frac{1}{b} + \frac{2}{b} - 2 \cdot \frac{1}{a} = \frac{3}{b} - \frac{2}{a} \]

### Step 4: Simplifying the Second Term
For the second term:

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

### Step 5: Multiply the Two Simplified Terms
Now we multiply the two simplified terms:

\[ \left( \frac{3}{b} - \frac{2}{a} \right) \left( 2 \cdot \frac{1}{a} - \frac{1}{b} \right) \]

Expanding this product:

\[ = \frac{3 \cdot 2}{ab} - \frac{3}{b^2} - \frac{4}{a^2} + \frac{2}{ab} \]

Combining like terms gives:

\[ = \frac{6}{ab} - \frac{3}{b^2} - \frac{4}{a^2} \]

### Step 6: Final Simplification
Now we need to express this in terms of a single fraction. To do this, we can find a common denominator, which is \(a^2b^2\):

\[ = \frac{6b - 3a^2 - 4b^2}{a^2b^2} \]

### Step 7: Conclusion
After simplifying, we can conclude that the expression evaluates to:

\[ \frac{4}{ac} - \frac{3}{b^2} \]

Thus, the final answer is:

\[ \boxed{\frac{4}{ac} - \frac{3}{b^2}} \]