The corner points of the feasible region for an L.P.P. are (0, 10), (5, 5), (5, 15) and (0, 30). If the objective function is $Z=\alpha x+\beta y, \alpha, \beta>0$, the condition on $\alpha$ and $\beta$ so that maximum of Z occurs at corner point (5, 5) and (0, 20) is:

$\alpha=3 \beta$

The correct answer is Option (3) → $\alpha=3 \beta$

$\text{Given corner points are }(0,10),(5,5),(5,15),(0,30)$

$Z=\alpha x+\beta y,\;\alpha>0,\beta>0$

$Z(5,5)=5\alpha+5\beta$

$Z(0,20)=20\beta$

$\text{For maximum at both points, }Z(5,5)=Z(0,20)$

$5\alpha+5\beta=20\beta$

$5\alpha=15\beta$

$\alpha=3\beta$

$\alpha=3\beta$