If a,b,c are in H.P, then

Options

A. `(a-b)/(b-c)=(a)/(c)`
B. `(b-c)/(c-a)=(b)/(a)`
C. `(c-a)/(a-b)=(c)/(b)`
D. `(a-b)/(b-c)=(c)/(a)`

Answer

To solve the problem, we need to establish the relationship between \(a\), \(b\), and \(c\) when they are 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.).

2. **Setting up the A.P. condition**:
For three numbers to be in A.P., the middle term must be the average of the other two. Thus, we can write:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]

3. **Rearranging the equation**:
This can be rearranged as:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]
Multiplying through by \(abc\) to eliminate the denominators gives:
\[ 2ac = bc + ab \]

4. **Rearranging further**:
Rearranging the equation gives:
\[ 2ac - ab - bc = 0 \]

5. **Finding the relationship**:
Rearranging this equation, we can express it in terms of the ratios:
\[ \frac{a - b}{b - c} = \frac{a}{c} \]

6. **Conclusion**:
Therefore, if \(a\), \(b\), and \(c\) are in H.P., we have established that:
\[ \frac{a - b}{b - c} = \frac{a}{c} \]

### Final Result:
The correct option based on the derived relationship is:
\[ \frac{a - b}{b - c} = \frac{a}{c} \]

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