<s>[INST] 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 $)   
 [/INST] Rearranging, we get $\sqrt{y} = 1 - \sqrt{x}$, which becomes $y = 1 - 2\sqrt{x} + x$ when we square both sides. Rearranging again and squaring both sides, we get $(y-x-1)^2 = y^2 + x^2 + 1 - 2xy - 2y + 2x = 4x$.
Generally, when there is an equation of the form $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$, if $B^2 - 4AC = 0$, then the curve is one of the following:


*

*A parabola

*2 Parallel Lines (certainly this is not the case)

*1 Line (certainly this is not the case)

*No curve (certainly this is not the case)


Therefore, $\sqrt{x} + \sqrt{y} = 1$ is a part of a parabola.
(Credits partly go to WyzAnt)
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