Question 4
Let $\alpha$ and $\beta$ be real numbers. Find, with proof, the maximal value of the expression
$$
|\alpha x + \beta y| + |\alpha x - \beta y|,
$$
in each of the following cases:
a) $x, y \in \mathbb{R}$, such that $|x| \le 1$ and $|y| \le 1$;
b) $x, y \in \mathbb{C}$, such that $|x| \le 1$ and $|y| \le 1$.
$$
|\alpha x + \beta y| + |\alpha x - \beta y|,
$$
in each of the following cases:
a) $x, y \in \mathbb{R}$, such that $|x| \le 1$ and $|y| \le 1$;
b) $x, y \in \mathbb{C}$, such that $|x| \le 1$ and $|y| \le 1$.