In a LPP, if the objective function Z=ax+by has the same maximum value of two corner points of the feasible region, then every point of the line segment joining these two points give the same ..value.

Explanation

To solve the given problem, we need to analyze the situation described in the context of Linear Programming Problems (LPP).

### Step-by-Step Solution:

1. **Understanding the Objective Function**:
The objective function is given as \( Z = ax + by \). This function represents a linear relationship between the variables \( x \) and \( y \).

**Hint**: Remember that the objective function is what we aim to maximize or minimize in a linear programming problem.

2. **Identifying Corner Points**:
In a feasible region defined by constraints, corner points (or vertices) are the points where the constraints intersect. The problem states that two corner points have the same maximum value for the objective function \( Z \).

**Hint**: Corner points are crucial in linear programming as they are potential candidates for the optimal solution.

3. **Analyzing the Condition**:
If two corner points, say \( P_1 \) and \( P_2 \), yield the same maximum value for \( Z \), we can denote this common value as \( Z_0 \). Thus, we have:
\[ Z(P_1) = Z_0 \quad \text{and} \quad Z(P_2) = Z_0 \]

**Hint**: This condition implies that the objective function is constant along the line segment connecting these two points.

4. **Considering the Line Segment**:
The line segment joining \( P_1 \) and \( P_2 \) consists of all points that can be expressed as:
\[ P(t) = (1-t)P_1 + tP_2 \quad \text{for } t \in [0, 1] \]
where \( P(t) \) represents any point on the line segment between \( P_1 \) and \( P_2 \).

**Hint**: The parameter \( t \) allows us to find any point on the line segment between the two corner points.

5. **Evaluating the Objective Function on the Segment**:
Since the objective function \( Z \) is linear, it will take the same value for any point \( P(t) \) on the line segment between \( P_1 \) and \( P_2 \). Therefore:
\[ Z(P(t)) = Z_0 \quad \text{for all } t \in [0, 1] \]

**Hint**: A linear function evaluated at points along a line segment between two points will yield the same result if those two points yield the same value.

6. **Conclusion**:
Hence, we can conclude that every point on the line segment joining the two corner points \( P_1 \) and \( P_2 \) gives the same **maximum value** of the objective function \( Z \).

**Final Answer**: The fill in the blank is "maximum".