If three real numbers a,b and c are in harmonic progression, then which of the folllowing is true ?

Options

A. ` 1/(bc),1/(ca),1/(ab)` are in HP
B. ab,bc,ca are inHP
C. ab,bc,ca are in HP
D. `a/b,b/c,c/a` are in HP

Explanation

To determine which statement is true when three real numbers \( a, b, c \) are in harmonic progression (HP), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Harmonic Progression**:
- Three numbers \( a, b, c \) are said to be in harmonic progression if the reciprocals of these numbers \( \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \) are in arithmetic progression (AP).

2. **Condition for AP**:
- For three numbers \( x, y, z \) to be in AP, the condition is:
\[ 2y = x + z \]
- Applying this to our reciprocals, we have:
\[ 2 \cdot \frac{1}{b} = \frac{1}{a} + \frac{1}{c} \]

3. **Rearranging the Equation**:
- Multiplying through by \( abc \) to eliminate the denominators gives:
\[ 2ac = bc + ab \]

4. **Identifying the True Statement**:
- From the equation \( 2ac = ab + bc \), we can manipulate it to find relationships among \( a, b, c \).
- Rearranging gives:
\[ 2ac - ab - bc = 0 \]
- This can be factored or analyzed to find specific relationships depending on the options provided.

5. **Conclusion**:
- The specific relationships derived from the equation will help us identify which of the provided options is true.