If `1/a, 1/b,1/c ` are in A.P., the `(1/a + 1/b - 1/c) ( 1/b+1/c-1/a)` is equal to

Options

A. `(4)/(ac)- (3)/(b^2)`
B. `(b^2 - ac)/(a^2 b^2 c^2)`
C. `(4)/(ac) - (1)/(b^2)`
D. none of these

Answer

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

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

From this, we can express \( \frac{1}{c} \) in terms of \( \frac{1}{a} \) and \( \frac{1}{b} \):

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

Next, we need to evaluate the expression \( \left( \frac{1}{a} + \frac{1}{b} - \frac{1}{c} \right) \left( \frac{1}{b} + \frac{1}{c} - \frac{1}{a} \right) \).

### Step 1: Simplifying the first part \( \frac{1}{a} + \frac{1}{b} - \frac{1}{c} \)

Substituting \( \frac{1}{c} \):

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

This simplifies to:

\[ 2 \cdot \frac{1}{a} - \frac{1}{b} \]

### Step 2: Simplifying the second part \( \frac{1}{b} + \frac{1}{c} - \frac{1}{a} \)

Again substituting \( \frac{1}{c} \):

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

This simplifies to:

\[ 3 \cdot \frac{1}{b} - 2 \cdot \frac{1}{a} \]

### Step 3: Multiplying the two simplified parts

Now we multiply the two results:

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

Expanding this product:

\[ = 2 \cdot \frac{1}{a} \cdot 3 \cdot \frac{1}{b} - 2 \cdot \frac{1}{a} \cdot \frac{1}{b} - \frac{1}{b} \cdot 3 \cdot \frac{1}{b} + \frac{1}{b} \cdot 2 \cdot \frac{1}{a} \]

This simplifies to:

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

Combining like terms:

\[ = \frac{8}{ab} - \frac{2}{a^2} - \frac{3}{b^2} \]

### Final Result

Thus, the final expression is:

\[ \frac{8}{ab} - \frac{2}{a^2} - \frac{3}{b^2} \]