Finding maximum value from a statement/equation The statement has values of $x$ and $y$ as positive integers: $$\sqrt{x} - \sqrt{11} = \sqrt{y}$$
I have to find the maximum possible values of $\frac{x}{y}$, this what I have done so far:
$$x = (\sqrt{y} + \sqrt{11})^2$$
$$y = (\sqrt{x} - \sqrt{11})^2$$
therefore: $$\frac{x}{y} = \frac{(\sqrt{y} + \sqrt{11})^2}{(\sqrt{x} - \sqrt{11})^2}$$
...
$$\frac{x}{y} = \frac{y + 11 + 2\sqrt{y}\sqrt{11}}{x + 11 - 2\sqrt{x}\sqrt{11}} = \frac{y+\sqrt{y}}{x-\sqrt{x}} + 1$$
and from here I'm not sure what to do...
(Again better title suggestions are also welcome)

Dividing by $\sqrt{y}$ we get 
$$\sqrt{\frac{x}{y}}=\sqrt{\frac{11}{y}}+1$$ 
and by squaring
$$\frac{x}{y}=\frac{11}{y}+2\sqrt{\frac{11}{y}}+1$$
So $\sqrt{\frac{11}{y}}$ should be rational and y is integer then for maximum we take y=11. so you get 
$\frac{x}{y}=4$
