For the given linear programming problem $z = ax + by, a,b>0$ subject to the constraints $2x + y ≤ 10,x+3y≤ 15,x,y ≥ 0$. If the corner points are (0, 0), (5, 0), (3, 4) and (0, 5) and $z$ is maximum at both (3, 4) and (0, 5), then the relationship between $a$ and $b$ is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $b=3a$
Given: $z = ax + by$, with $a, b > 0$
Corner points: $(0, 0)$, $(5, 0)$, $(3, 4)$, $(0, 5)$
It is given that $z$ attains maximum at both $(3, 4)$ and $(0, 5)$
So, $z(3, 4) = z(0, 5)$
$a \cdot 3 + b \cdot 4 = a \cdot 0 + b \cdot 5$
$3a + 4b = 5b$
$3a = b$