The corners points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z= ax+ by, where a, b > 0. Condition on a and b so that the maximizing value of Z occurs at both the points (15, 15) and (0, 20) is :

b = 3a

Given objective function

$Z=ax+by,\;a,b>0$

Maximum occurs at both points $(15,15)$ and $(0,20)$

So

$Z(15,15)=Z(0,20)$

$15a+15b=20b$

$15a=5b$

$b=3a$

The required condition is $b=3a$.