Prove $x=-y$: A Math Challenge

[ipaper]

Let $x, y$ be real numbers such that
$$(\sqrt{y^{2} - x\,\,}\, - x)(\sqrt{x^{2} + y\,\,}\, - y)=y.$$
Prove $x=-y$.

Any suggestion would be appreciated.

 

[Kiwi1]

Another way of stating the question is:

Show that x=-y is a solution to:

$$\left[\sqrt{y^2-x}-x \right]\left[\sqrt{x^2+y}-y\right]=y$$

Writing it is this way it is more obvious that all you have to do is substitute (-y) for each of the x's in the left hand side and then simplify to show that the left hand side is equal to y.

To do this it will be helpful to remember the identity:
$$(a+b)(a-b)=(a^2-b^2)$$

 

[Evgeny.Makarov]

Another way of stating the question is:

Show that x=-y is a solution to:

$$\left[\sqrt{y^2-x}-x \right]\left[\sqrt{x^2+y}-y\right]=y$$

No, this is the converse of what the original question is asking.