Find sides of a right triangle given hypotenuse c and area A (no numbers given)

Question

I've solved couple of these, but I have no idea how to solve it without any numbers provided. I've tried using $A=\frac{ab}{2} \Rightarrow 2A=ab \Rightarrow 4A^2=a^2b^2$ and incorporating $b^2=c^2-a^2$, but I don't know what to do next...

$$c^2=a^2+b^2$$ $$b^2=c^2-a^2$$

Then we proceed as:

$$A=\frac{ab}{2}$$ $$2A=ab$$ $$4A^2=a^2b^2$$ $$4A^2=a^2(c^2-a^2)$$ $$4A^2=a^2c^2-a^4$$ $$a^4-a^2c^2+4A^2=0$$ ... what do I do next?

Answer 1

You already noted $A = \frac{ab}{2} $ and $a^2+b^2 = c^2$. Now note $ab = 2A$ and $$(a+b)^2 = a^2+2ab+b^2 = c^2+4A$$

$$\Rightarrow a+b = \sqrt{c^2+4A}$$

Can you go from here?