Let the corner points of the bounded feasible region of the linear programming problem (LPP) $Z = ax+by$ be (0, 0), (2, 0), (20/19, 45/19) and (0, 3). if the optimal value of Z occurs at both points (2, 0) and (20/19, 45/19), then the relation between $a$ and $b$ is:

$2a=5b$

The correct answer is Option (3) → $2a=5b$

Given LPP with $Z = a x + b y$ and corner points: $(0,0), (2,0), (20/19, 45/19), (0,3)$

Optimal value occurs at both $(2,0)$ and $(20/19, 45/19)$, so:

$Z$ at $(2,0) = Z$ at $(20/19, 45/19)$

$2a + 0b = \frac{20}{19}a + \frac{45}{19}b$

Multiply both sides by 19: $38a = 20a + 45b$

$38a - 20a = 45b \Rightarrow 18a = 45b$

$\frac{a}{b} = \frac{45}{18} = \frac{5}{2}$

Relation: $a : b = 5 : 2$