The corner points of the bounded feasible region of an LPP are O(0,0), A(250,0), B(200,50) and C(0,175) as given in the figure below. If the maximum value of the objective function \( z=2ax+by \) occur at the points A(250, 0) and B(200, 50), then the relation between \( a \) and \( b \) is :
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 2a=b \)
If the maximum occurs at two corner points, z must have the same value at both of them. The four corner points are given in the question, so no figure is needed.
\( \displaystyle z(\mathrm{A})=2a(250)+b(0)=500a \)
\( \displaystyle z(\mathrm{B})=2a(200)+b(50)=400a+50b \)
\( \displaystyle 500a=400a+50b \)
\( \displaystyle 100a=50b \)
\( \displaystyle 2a=b \)
Shortcut: The maximum occurs at every point of the edge AB, so the line \( 2ax+by=k \) is parallel to AB.
\( \displaystyle \text{slope of AB}=\frac{50-0}{200-250}=-1 \)
\( \displaystyle \text{slope of }2ax+by=k\ \text{is}\ -\frac{2a}{b} \)
\( \displaystyle -\frac{2a}{b}=-1\ \Rightarrow\ 2a=b \)
NCERT Exemplar:
Linear Programming
