Solve the linear program graphically to find the optimal solution. $$\text{max}~2x_1$$ such that $$x_1+x_2\leq 10$$ $$2x_1-x_2 \leq 5$$ $$x_1,x_2 \geq 0$$
Now, for this part, it is relatively to find the optimal point - answer is just their intersection point $(5,5)$.
Now the next part, the question asked, suppose we change the objective function to $2x_1+ax_2$, what is the range of $a$ such that the optimal solution in the previous part remains optimal.
I am having trouble fully understanding the answer: The normal vectors for the active constraints are $(1,1),(2,-1)$. Scale accordingly $(1,1)$ to $(2,2)$. Now the normal vector for the objective function is $(2,a)$. For the normal vector $(2,a)$ to be in the convex hull of $(2,2)$ and $(2,-1)$, we have the range $-1 \leq a \leq 2$ as needed.
What i do not understand is what has convex hull got to do with this question (only briefly know the definition of convex hull in my lecture notes). And is there any other way to solve this question easily (comparing gradient of objective functions?)