The corner points of the feasible region determined by

x + y ≤ 8, 2x + y ≥ 8, x ≥ 0, y ≥ 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:

a = 2b

The correct answer is Option (2) → a = 2b

Points A(0, 8) and B(4, 0) form the line segment AB.

Slope of AB = (0 − 8) / (4 − 0) = −8/4 = −2

For the objective function Z=ax+by

Its lines are of the form ax+by=k whose slope is −a/b.

For maximum value to lie along AB, the objective line must be parallel to AB, so:
−a/b = −2
⇒ a/b = 2
⇒ a = 2b

The correct answer is Option (2) → a = 2b