The corner points of the bounded feasible region determined by the system of linear constraints are : (20, 0), (3, 0), (0, 7), (2,3) and (5, 5). If the objective function is \( z=ax+by \) , where \( a,b\gt 0 \) ,then the condition on \( a \) and \( b \), so that the minimum of \( z \) occurs at (20,0) and (5,5) is,
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 3a=b \)
If the minimum occurs at both (20, 0) and (5, 5), the objective function has the same value at the two points:
\( 20a=5a+5b \)
\( 15a=5b \)
\( b=3a \)
Note: It seems that there is a mistake in the original NTA paper, which prints corner points that do not fit a minimum at (20, 0) and (5, 5). The corner (3, 0) gives \( z=3a \), which is smaller than \( z=20a \) at (20, 0) for every \( a\gt 0 \); and (2, 3) lies inside the region formed by the other four points, so it cannot be a corner point. The question can still be solved, because the condition comes directly from equating the values of z at the two points named in the question, and only Option (4) gives it.