Let $a,b \in \mathbb R$, $R \ge 0$, and $c > 0$. Define $C := \{(x,y) \in \mathbb R^d \mid x^2 + y^2 \le 1,\,x^2 + c y^2 \le R^2\}$, and set

$$ \alpha := \sup_{(x,y) \in C} ax + b y. $$

Question. In terms of $a,b,c,R$, is there an analytic formula for $\alpha$ ?

Some special cases

• If $c=1$, then $C = \{z \in \mathbb R^2 \mid \|z\|_2 \le R'\}$, where $R' := \min(1,R)$. Thus, $\alpha = R'\sqrt{a^2+b^2}$.
• If $a=0$, then we are maximizing $yb$ over $[-1,1] \cap [-R/\sqrt c,R/\sqrt c]$, and so $\alpha = |a|\min(1,R/\sqrt c)$.
• $\lim_{c \to \infty} \alpha = |a|R'$. This is because the domain $C$ is shrunk to the interval $[-R',R'] \times \{0\}$.