Three cities Fatehpur, Barabanki and Lucknow are famous. Fatehpur is 42 km, away from Barakanki and the distance betweén Fatehpur and Lucknow is 66 km. Which of the following cannot be the distance between Barabanki and Lucknow?

Options

A. A)28 km
B. B)98 km
C. C)32 km
D. D)23 km

Answer

To solve the problem, we need to determine which distance between Barabanki and Lucknow cannot be possible given the distances between the other two cities, Fatehpur and Barabanki, and Fatehpur and Lucknow.

### Step-by-Step Solution:

1. **Identify the distances given:**
- Distance from Fatehpur to Barabanki (FB) = 42 km
- Distance from Fatehpur to Lucknow (FL) = 66 km

2. **Let the distance from Barabanki to Lucknow (BL) be represented as K.**

3. **Apply the Triangle Inequality Theorem:**
The triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we can set up the following inequalities:
- FB + FL > BL
- FB + BL > FL
- FL + BL > FB

4. **Substituting the known distances into the inequalities:**
- From FB + FL > BL:
\[ 42 + 66 > K \implies 108 > K \implies K < 108 \]
- From FB + BL > FL:
\[ 42 + K > 66 \implies K > 66 - 42 \implies K > 24 \]
- From FL + BL > FB:
\[ 66 + K > 42 \implies K > 42 - 66 \implies K > -24 \quad (\text{This inequality is always true since K is positive}) \]

5. **Combine the results:**
From the inequalities derived, we find:
\[ 24 < K < 108 \]
This means that the distance K (Barabanki to Lucknow) must be greater than 24 km and less than 108 km.

6. **Evaluate the options:**
Now, we need to check the given options to see which one does not lie within the range (24, 108):
- Option A: 28 km (within range)
- Option B: 98 km (within range)
- Option C: 32 km (within range)
- Option D: 23 km (not within range)

7. **Conclusion:**
The distance that cannot be the distance between Barabanki and Lucknow is **23 km**.