The amphoteric hydroxide is:

First, let us recall the definition of an amphoteric hydroxide. A hydroxide is called amphoteric if it has the dual ability to react both with acids, behaving as a base, and with strong bases, behaving as an acid. Mathematically, we can symbolize these two kinds of reactions as

$$\text{Amph. OH} + \text{acid} \;\longrightarrow\; \text{salt} + H_2O$$

and

$$\text{Amph. OH} + \text{strong base} \;\longrightarrow\; \text{complex anion (salt)} + H_2O.$$

Now we test each hydroxide given in the options against this criterion.

Testing $$Be(OH)_2$$. When we add hydrochloric acid, $$Be(OH)_2$$ dissolves by acting as a base:

$$Be(OH)_2 + 2HCl \;\longrightarrow\; BeCl_2 + 2H_2O.$$

On the other hand, when we add a strong base such as sodium hydroxide, $$Be(OH)_2$$ behaves as an acid and forms a soluble beryllate complex:

$$Be(OH)_2 + 2NaOH \;\longrightarrow\; Na_2BeO_2 + 2H_2O.$$

Because it reacts in both directions, $$Be(OH)_2$$ is amphoteric.

Testing $$Ca(OH)_2$$, $$Mg(OH)_2$$, and $$Sr(OH)_2$$. Each of these hydroxides is clearly basic. They react readily with acids, for example

$$Ca(OH)_2 + 2HCl \;\longrightarrow\; CaCl_2 + 2H_2O,$$

but when treated with excess strong base they do not form any additional soluble complex anion; instead they simply remain undissolved or precipitate. Thus they are not amphoteric; they are purely basic hydroxides.

Since only $$Be(OH)_2$$ satisfies the amphoteric criterion, we conclude that it is the required hydroxide.

Hence, the correct answer is Option A.