The corner points of the feasible region determined by a system of linear inequations are (0,0),(4,0),(2,4) and (0,5). If the minimum value of Z=ax+by,a, `bgt0` occurs at (2,4) and (0,5) ,then :

Options

A. `a=2b`
B. `2a=b`
C. `a=b`
D. `3a=b`

Answer

To solve the problem, we need to find the relationship between the coefficients \( A \) and \( B \) in the objective function \( Z = ax + by \) given that the minimum value occurs at the points (2, 4) and (0, 5).

### Step-by-step Solution:

1. **Substitute the point (2, 4) into the objective function**:
\[ Z(2, 4) = a(2) + b(4) = 2a + 4b \]

2. **Substitute the point (0, 5) into the objective function**:
\[ Z(0, 5) = a(0) + b(5) = 5b \]

3. **Set the two expressions for Z equal to each other** since both points yield the minimum value:
\[ 2a + 4b = 5b \]

4. **Rearrange the equation** to isolate \( a \):
\[ 2a + 4b - 5b = 0 \]
\[ 2a - b = 0 \]

5. **Solve for \( b \)** in terms of \( a \):
\[ 2a = b \]

6. **Express the relationship**:
\[ b = 2a \]

### Conclusion:
From the derived relationship \( b = 2a \), we can see that option 2, which states \( 2A = B \), is the correct answer.