Two vessels P and Q contain 'a' litres of petrol and 'b' litres of kerosene respectively. 'c' litres of petrol and same quantity of kerosene is taken out and then transferred to Q and P respectively. This process is repeated several times. If after the first operation the quantity of petrol or kerosene in either of P and Q does not change. What is the value of 'c'?

Options

A. (A) `(ab)/((a-b))`
B. (B) `(2ab)/((a+b))`
C. (C) `(ab)/((a+b))`
D. (D) `(a/b)^2`

Answer

To solve the problem, we need to determine the value of 'c' such that after transferring 'c' litres of petrol from vessel P to vessel Q and 'c' litres of kerosene from vessel Q to vessel P, the quantities of petrol in vessel P and kerosene in vessel Q remain unchanged after several iterations of this process.

### Step-by-Step Solution:

1. **Initial Conditions**:
- Vessel P contains 'a' litres of petrol.
- Vessel Q contains 'b' litres of kerosene.

2. **First Transfer**:
- We take out 'c' litres of petrol from vessel P.
- We also take out 'c' litres of kerosene from vessel Q.
- After this transfer:
- Vessel P will have \( a - c \) litres of petrol.
- Vessel Q will have \( b - c \) litres of kerosene.

3. **Second Transfer**:
- We now transfer 'c' litres of kerosene from vessel Q back to vessel P.
- We also transfer 'c' litres of petrol from vessel P back to vessel Q.
- After this transfer:
- Vessel P will have \( (a - c) + c = a \) litres of petrol.
- Vessel Q will have \( (b - c) + c = b \) litres of kerosene.

4. **Equilibrium Condition**:
- The problem states that after several iterations, the quantities of petrol in vessel P and kerosene in vessel Q do not change. This implies that the amount of petrol and kerosene being transferred must balance out perfectly.

5. **Deriving the Formula**:
- For the quantities to remain unchanged, the amount of petrol transferred must equal the amount of kerosene transferred in terms of their respective volumes.
- This leads us to the equation:
\[ \frac{c}{a} = \frac{c}{b} \]
- Rearranging gives us:
\[ c = \frac{ab}{a + b} \]

6. **Final Result**:
- Therefore, the value of 'c' that ensures the quantities do not change after repeated operations is:
\[ c = \frac{ab}{a + b} \]