The corner points of the bounded feasible region determined by a set of constraints (linear inequalities) are A(0, 5), B(3, 5), C(5, 0) and D(4, 1) and the objective function is $z = px + 2qy$ where $p, q> 0$. The condition on $p$ and $q$, such that, the maximum $z$ occurs at B and D, is

$p-8q=0$

The correct answer is Option (1) → $p-8q=0$

$z_A = 0 \cdot p + 5 \cdot (2q) = 10q$

$z_B = 3p + 5 \cdot (2q) = 3p + 10q$

$z_C = 5p + 0 = 5p$

$z_D = 4p + 1 \cdot (2q) = 4p + 2q$

For maxima at B and D:

$z_B = z_D \ \Rightarrow \ 3p + 10q = 4p + 2q$

$\Rightarrow \ p - 8q = 0$