Parabola $\sqrt {x}+\sqrt {y}=1 $ How do I prove that the equation $\sqrt {x}+\sqrt {y}=1 $ is part of parabola. 
My attempt:rotation in 45 degrees brings the equation to $ -2a^2=1-2\sqrt {2}b $ when $ x= \frac {a-b} {\sqrt {2} } $ and $ y= \frac {a+b} {\sqrt {2} } $. It is a parabola, why is it only part of it? (also for $\sqrt {x}-\sqrt {y}=1 $)   

Square both sides:
$$x + 2 \sqrt{xy} + y = 1.$$
Isolate the radical and square:
$$xy = \left(\frac{1-x-y}{2}\right)^2.$$
Rotate the axes by $\pi/4$ to new axes ${X,Y}$:
$$\frac{Y^2-X^2}{2} = \left(\frac{1 - \sqrt{2}Y}{2}\right)^2.$$
Simplfying gives:
$$X^2 = 2\sqrt{2}Y - 1,$$
which is the equation for a parabola.
