If a,b,c are in H.P, where `a gt c gt 0,` then :

Options

A. `b gt (a+c)/(2)`
B. `(1)/(a-b) -(1)/(b-c) lt 0`
C. `ac gt b ^(2)`
D. `bc (1-a), ac (1-b) , ab (1- c)` are in A.P.

Answer

To solve the problem, we need to analyze the given information that \( a, b, c \) are in Harmonic Progression (H.P.) and \( a > c > 0 \).

### Step-by-Step Solution:

1. **Understanding H.P.**:
If \( a, b, c \) are in H.P., then their reciprocals \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in Arithmetic Progression (A.P.).

2. **Setting Up the A.P. Condition**:
For \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) to be in A.P., the following condition must hold:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
Rearranging gives:
\[ \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \]

3. **Finding a Common Denominator**:
To combine the right-hand side, we can find a common denominator:
\[ \frac{1}{a} + \frac{1}{c} = \frac{c + a}{ac} \]
Thus, we have:
\[ \frac{2}{b} = \frac{c + a}{ac} \]

4. **Cross-Multiplying**:
Cross-multiplying gives:
\[ 2ac = b(c + a) \]
This equation relates \( a, b, c \).

5. **Verifying the Options**:
We need to check the given options based on the relationships derived above.

- **Option 1**: Check if \( b > \frac{a+c}{2} \).
- From our earlier equation, we can substitute values to check if this holds true.

- **Option 2**: Check if \( \frac{1}{a} - \frac{1}{b} < \frac{1}{b} - \frac{1}{c} \).
- Substitute \( a, b, c \) into this inequality to verify.

- **Option 3**: Check if \( ac > b^2 \).
- Substitute \( a, b, c \) values to check this inequality.

- **Option 4**: Check if \( bc(1-a) = ab(1-c) \).
- Substitute values and verify if this holds true.

6. **Conclusion**:
After checking all options, we conclude which options are correct based on the inequalities and relationships derived from the H.P. condition.