If `1/a+1/(a-2b)+1/c+1/(c-2b)=0 and a,b,c` are not in A.P., then

Options

A. a,b,c are in G.P
B. a,b/2,x are in A.P
C. a,b/2,c are in H.P
D. a,2b,c are in H.P

Answer

To solve the problem, we need to analyze the equation given and the conditions provided. The equation is:

\[ \frac{1}{a} + \frac{1}{a - 2b} + \frac{1}{c} + \frac{1}{c - 2b} = 0 \]

We also know that \(a\), \(b\), and \(c\) are not in arithmetic progression (A.P.).

### Step 1: Rearranging the Equation
We can rearrange the equation by grouping the terms:

\[ \left( \frac{1}{a} + \frac{1}{c} \right) + \left( \frac{1}{a - 2b} + \frac{1}{c - 2b} \right) = 0 \]

### Step 2: Finding a Common Denominator
Next, we can find a common denominator for the first group and the second group:

\[ \frac{c + a}{ac} + \frac{(c - 2b) + (a - 2b)}{(a - 2b)(c - 2b)} = 0 \]

### Step 3: Simplifying the Second Group
The second group simplifies to:

\[ \frac{(a + c - 4b)}{(a - 2b)(c - 2b)} = 0 \]

### Step 4: Setting the Numerator to Zero
For the entire expression to equal zero, the numerator must equal zero (since the denominators cannot be zero):

\[ a + c - 4b = 0 \]

### Step 5: Rearranging the Equation
Rearranging gives us:

\[ a + c = 4b \]

### Step 6: Analyzing the Condition
Since \(a\), \(b\), and \(c\) are not in arithmetic progression, we need to check the implications of \(a + c = 4b\). This suggests that \(b\) is the average of \(a\) and \(c\), which contradicts the condition that they are not in A.P.

### Step 7: Establishing Harmonic Progression
From \(a + c = 4b\), we can express this in terms of harmonic progression. The condition can be rewritten as:

\[ \frac{1}{a} + \frac{1}{c} = \frac{2}{2b} \]

This indicates that \(a\), \(2b\), and \(c\) are in harmonic progression (H.P.).

### Conclusion
Thus, we conclude that:

\[ a, 2b, c \text{ are in HP.} \]

### Final Answer
The correct option is that \(a, 2b, c\) are in harmonic progression.

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