Let $a,b\in\mathbb{R}$ not both $0$ and $f(\vec{x})=ax_1+bx_2$ and $A= \{f(\vec{x}):\vec{x}\in B(\vec{0},1)\}$. Determine $\inf A$ and $\sup A$ where $$B(\vec{0},1)=\{\vec{y}=(y_1,y_2)\in\mathbb{R}^2: \sqrt{y_1^2+y_2^2}<1\}$$ and $\inf, \sup$ denote the infimum and supremum respectively.
By intuition, I think that the supremum and minimum would be the last and first intersection of the line $bx_1=ax_2$ with the circle of radius $1$ centered on $0,0$ respectively, but I don't know how to prove this or how to start to be honest. I will appreciate any help to solve this. Thanks in advance for reading