Question:

The corner points of the feasible region determined by $x + y \leq 8$, $2x + y \geq 8$, $x \geq 0$, $y \geq 0$ are $A(0, 8)$, $B(4, 0)$, and $C(8, 0)$. If the objective function $Z = ax + by$ has its maximum value on the line segment $AB$, then the relation between $a$ and $b$ is:

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When dealing with linear optimization problems, the slope of the constraint line often provides a critical relationship between the coefficients of the objective function. Make sure to relate the slope of the constraint line to the ratio of coefficients for finding optimal values.

Updated On: Mar 30, 2025

$8a + 4 = b$
$a = 2b$
$b = 2a$
$8b + 4 = a$

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The Correct Option is B

Approach Solution - 1

The line segment $AB$ has the points $A(0, 8)$ and $B(4, 0)$. The objective function $Z = ax + by$ will have a maximum value on $AB$ if $\frac{a}{b} = -\frac{\text{change in } y}{\text{change in } x}$.

Between points $A$ and $B$:

Slope of $AB$ is given by:

\[\text{Slope of } AB = \frac{0 - 8}{4 - 0} = -2\]

Thus, the ratio $\frac{a}{b} = 2$ implies $a = 2b$.

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Approach Solution -2

Between points $A$ and $B$:

Step 1: Calculate the slope of the line segment $AB$.

The formula for the slope of a line is given by:

\[ \text{Slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} \] where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the points $A$ and $B$. For the points $A(0, 8)$ and $B(4, 0)$, we have: \[ \text{Slope of } AB = \frac{0 - 8}{4 - 0} = \frac{-8}{4} = -2 \]

Step 2: Relating the slope to the objective function.

The objective function $Z = ax + by$ has a maximum value when the ratio $\frac{a}{b}$ matches the negative of the slope of the line segment $AB$, i.e., \[ \frac{a}{b} = -\text{Slope of } AB \] Substituting the value of the slope: \[ \frac{a}{b} = -(-2) = 2 \]

Step 3: Expressing the relationship between $a$ and $b$.

From the equation $\frac{a}{b} = 2$, we can express $a$ in terms of $b$: \[ a = 2b \]

Conclusion: The objective function will have a maximum value on the line segment $AB$ if $a = 2b$.