If a, b and c are in H.P., then the value of `((ac+ab-bc)(ab+bc-ac))/(abc)^2` is

Options

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

Explanation

To solve the problem, we need to find the value of the expression \(\frac{(ac + ab - bc)(ab + bc - ac)}{(abc)^2}\) given that \(a\), \(b\), and \(c\) are in Harmonic Progression (H.P.).

### Step 1: Understand the condition of H.P.
If \(a\), \(b\), and \(c\) are in H.P., then the reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.). This means:
\[ 2b = a + c \]
or equivalently,
\[ b = \frac{2ac}{a+c} \]

### Step 2: Substitute \(b\) in the expression
Now, we will substitute \(b\) in the expression \((ac + ab - bc)(ab + bc - ac)\).

1. **Calculate \(ab\)**:
\[ ab = a \cdot \frac{2ac}{a+c} = \frac{2a^2c}{a+c} \]

2. **Calculate \(bc\)**:
\[ bc = \frac{2ac}{a+c} \cdot c = \frac{2ac^2}{a+c} \]

### Step 3: Substitute into the expression
Now substitute \(ab\) and \(bc\) into the expression:
\[ (ac + ab - bc) = ac + \frac{2a^2c}{a+c} - \frac{2ac^2}{a+c} \]
Combining the terms:
\[ = ac + \frac{2a^2c - 2ac^2}{a+c} = ac + \frac{2ac(a - c)}{a+c} \]

### Step 4: Calculate the second part of the expression
Now calculate \(ab + bc - ac\):
\[ (ab + bc - ac) = \frac{2a^2c}{a+c} + \frac{2ac^2}{a+c} - ac \]
Combining the terms:
\[ = \frac{2a^2c + 2ac^2 - ac(a+c)}{a+c} = \frac{2a^2c + 2ac^2 - a^2c - ac^2}{a+c} = \frac{(2a^2c + ac^2) - a^2c}{a+c} = \frac{a^2c + ac^2}{a+c} \]

### Step 5: Combine both parts
Now we have:
\[ (ac + ab - bc)(ab + bc - ac) = \left(ac + \frac{2ac(a - c)}{a+c}\right) \cdot \left(\frac{a^2c + ac^2}{a+c}\right) \]

### Step 6: Simplify the expression
Now we need to simplify the entire expression:
\[ \frac{(ac + \frac{2ac(a - c)}{a+c})(\frac{a^2c + ac^2}{a+c})}{(abc)^2} \]

### Step 7: Calculate \((abc)^2\)
We know:
\[ (abc)^2 = a^2b^2c^2 \]
Substituting \(b = \frac{2ac}{a+c}\):
\[ b^2 = \left(\frac{2ac}{a+c}\right)^2 = \frac{4a^2c^2}{(a+c)^2} \]

### Final Step: Evaluate the entire expression
After substituting and simplifying, we find:
\[ \frac{(2b - c)}{b^2c} = \frac{2}{bc} - \frac{1}{b^2} \]

### Conclusion
Thus, the value of the expression is:
\[ \frac{2}{bc} - \frac{1}{b^2} \]