If three positive numbers `a, b, c` are in `A.P. and 1/a^2,1/b^2,1/c^2` also in `A.P.` then

Options

A. ` a = b = c `
B. `2b = 3a + c`
C. `b^(2) = (ac)/(8)`
D. `2c = 2b + a`

Answer

To solve the problem, we need to analyze the conditions given for the three positive numbers \( a, b, c \) being in arithmetic progression (A.P.) and the reciprocals of their squares \( \frac{1}{a^2}, \frac{1}{b^2}, \frac{1}{c^2} \) also being in A.P.

### Step-by-Step Solution:

1. **Understanding A.P.**:
If \( a, b, c \) are in A.P., then the condition can be expressed as:
\[ 2b = a + c \]

2. **Reciprocals in A.P.**:
For the numbers \( \frac{1}{a^2}, \frac{1}{b^2}, \frac{1}{c^2} \) to be in A.P., we have:
\[ 2 \cdot \frac{1}{b^2} = \frac{1}{a^2} + \frac{1}{c^2} \]
Rearranging gives us:
\[ \frac{1}{b^2} - \frac{1}{a^2} = \frac{1}{c^2} - \frac{1}{b^2} \]

3. **Cross Multiplying**:
We can rewrite the above equation using a common denominator:
\[ \frac{a^2 - b^2}{a^2 b^2} = \frac{b^2 - c^2}{b^2 c^2} \]

4. **Cross Multiplying Again**:
Cross-multiplying gives:
\[ (a^2 - b^2) c^2 = (b^2 - c^2) a^2 \]

5. **Expanding Both Sides**:
Expanding both sides results in:
\[ a^2 c^2 - b^2 c^2 = b^2 a^2 - c^2 a^2 \]

6. **Rearranging Terms**:
Rearranging the equation gives:
\[ a^2 c^2 - b^2 a^2 + c^2 b^2 - b^2 c^2 = 0 \]
This can be factored as:
\[ c^2 (a^2 - b^2) + b^2 (c^2 - a^2) = 0 \]

7. **Factoring Further**:
Factoring out common terms leads us to:
\[ (c - a)(c + a) + b^2 (c - a) = 0 \]
This implies:
\[ (c - a)(c + a + b^2) = 0 \]

8. **Solving the Factors**:
From the above equation, we have two cases:
- \( c - a = 0 \) which implies \( c = a \)
- \( c + a + b^2 = 0 \) which is not possible since \( a, b, c \) are positive.

9. **Conclusion**:
Therefore, since \( c = a \) and \( 2b = a + c \), we conclude:
\[ a = b = c \]

### Final Result:
Thus, the solution is:
\[ \boxed{a = b = c} \]