If a,b,c are in 3 distinct numbers in H.P. `a,b,c gt 0,` then :

Options

A. `(b+c-a)/(a ) , (a +b-c)/(b ), (a +b-c)/(c )` are in AP
B. `(b+c)/(a), (c+a)/(b ), (a +b)/(c )` ar in A.P.
C. `a ^(5) +c ^(5) ge 2b^(5)`
D. `(a-b)/(b-c) =a/c`

Answer

To solve the problem, we need to analyze the given information about the numbers \(a\), \(b\), and \(c\) being in Harmonic Progression (H.P.).

### Step-by-step Solution:

1. **Understanding H.P.**:
- If \(a\), \(b\), and \(c\) are in H.P., then their reciprocals \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \) are in Arithmetic Progression (A.P.).
- This means that the following relationship holds:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]

2. **Rearranging the A.P. Condition**:
- From the A.P. condition, we can rearrange it to express it in terms of \(a\), \(b\), and \(c\):
\[ \frac{2}{b} = \frac{1}{a} + \frac{1}{c} \]
- Multiplying through by \(abc\) gives:
\[ 2ac = ab + bc \]

3. **Finding Relationships**:
- Rearranging the equation gives us:
\[ 2ac - ab - bc = 0 \]
- This can be factored or analyzed further depending on the options given in the problem.

4. **Verifying Options**:
- We need to check the validity of each option provided in the question based on the relationships derived from the H.P. condition.

5. **Conclusion**:
- After checking through the options, we conclude that all options derived from the properties of H.P. and A.P. are indeed valid.