Consider the reaction :-
`2CO(g)+2H_(2)O_((g))hArr2CO_(2(g))+2H_(2(g))eq.const=K_(1)`
`CH_(4(g))+H_(2)O_((g))hArrCO_((g))+3H_(2(g)),eq.const=K_(2)`
`CH_(4(g))+2H_(2)O_((g))hArrCO_(2(g))+4H_(2(g)),eq.const=K_(3)`
Which of the following relation is correct ?

Options

A. `K_(3)=(K_(1))/(K_(2))`
B. `K_(3)=(K_(1)^(2))/(K_(2)^(2))`
C. `K_(3)=K_(1)K_(2)`
D. `K_(3)=sqrt(K_(1)).K_(2)`

Answer

To solve the problem, we need to analyze the given chemical reactions and their equilibrium constants to derive a relationship between them.

### Step-by-Step Solution:

1. **Identify the Reactions and Their Equilibrium Constants**:
- Reaction 1:
\[ 2 \text{CO}(g) + 2 \text{H}_2\text{O}(g) \rightleftharpoons 2 \text{CO}_2(g) + 2 \text{H}_2(g) \quad (K_1) \]
- Reaction 2:
\[ \text{CH}_4(g) + \text{H}_2\text{O}(g) \rightleftharpoons \text{CO}(g) + 3 \text{H}_2(g) \quad (K_2) \]
- Reaction 3:
\[ \text{CH}_4(g) + 2 \text{H}_2\text{O}(g) \rightleftharpoons \text{CO}_2(g) + 4 \text{H}_2(g) \quad (K_3) \]

2. **Manipulate Reaction 1**:
- Divide Reaction 1 by 2:
\[ \text{CO}(g) + \text{H}_2\text{O}(g) \rightleftharpoons \text{CO}_2(g) + \text{H}_2(g) \]
- The equilibrium constant for this modified reaction becomes:
\[ K' = K_1^{1/2} = \sqrt{K_1} \]

3. **Add the Modified Reaction to Reaction 2**:
- Now, we add the modified Reaction 1 to Reaction 2:
\[ \text{CO}(g) + \text{H}_2\text{O}(g) \rightleftharpoons \text{CO}_2(g) + \text{H}_2(g) \quad (K' = \sqrt{K_1}) \]
\[ \text{CH}_4(g) + \text{H}_2\text{O}(g) \rightleftharpoons \text{CO}(g) + 3 \text{H}_2(g) \quad (K_2) \]

4. **Combine the Reactions**:
- When we add these two reactions, the CO on the reactant side of the second reaction cancels with the CO on the product side of the first reaction:
\[ \text{CH}_4(g) + 2 \text{H}_2\text{O}(g) \rightleftharpoons \text{CO}_2(g) + 4 \text{H}_2(g) \]
- The resultant reaction is Reaction 3.

5. **Determine the Overall Equilibrium Constant**:
- The equilibrium constant for the overall reaction (Reaction 3) is the product of the equilibrium constants of the individual reactions:
\[ K_3 = K' \cdot K_2 = \sqrt{K_1} \cdot K_2 \]

6. **Final Relation**:
- Therefore, we can express the relationship as:
\[ K_3 = \sqrt{K_1} \cdot K_2 \]

### Conclusion:
The correct relation among the equilibrium constants is:
\[ K_3 = \sqrt{K_1} \cdot K_2 \]