reference on $\sqrt{ax}+\sqrt{by}=c$ as a parabola? Does anyone have a reference on the equation
$$\sqrt{ax}\,+\sqrt{by}=c\ ?$$
Clearing square roots and rearranging gives
$$ax+by = \frac{(ax-by)^2+c^4}{2c^2}$$
This is the equation of a parabola, so the original equation is a parabolic arc.  
I'm surprised because I've never seen this before:  I've never seen such a short equation yield a parabola at an angle from the coordinate axes.  Does anyone have any reference discussing this?
 A: no reference... I cannot tell whether you know how to do this. EDIT: judging from your MO answers, you do. Would have been better to include this in your question.... 
Rotated coordinate system by
$$ u = \frac{ax-by}{\sqrt {a^2 + b^2}},  $$
$$ v = \frac{bx+ay}{\sqrt {a^2 + b^2}},  $$
$$ x = \frac{au+bv}{\sqrt {a^2 + b^2}},  $$
$$ y = \frac{-bu+av}{\sqrt {a^2 + b^2}}.  $$
In particular
$$ ax+by = \frac{(a^2 - b^2)u + 2abv}{\sqrt {a^2 + b^2}}.  $$
Let us add in
$$ ax-by = \left(\sqrt {a^2 + b^2}\right) u.  $$
Taking your final formula, I get
$$ (a^2-b^2)u  + 2ab v = \frac{\sqrt {a^2 + b^2}}{2c^2} \left( (a^2 + b^2)u^2 + c^4 \right) $$
and one may solve for $v$ as some
$$  v = E u^2 + F u + G $$
which is a parabola, as advertised. The original solution set is the subset of this that is close to the origina and stops at the two points where the parabola is tangent to the $x$ and $y$ axes. 
Lat night, I forgot one factor of $2,$ which is why I got the wrong conclusion. It became much clearer after I did the symmetric example $\sqrt x + \sqrt y = 1,$ which is rotated exactly $45^\circ.$ So, I suggest including one or two examples, typeset, so as to offset the influence of too many symbols.
A: Once the equation 
$$ \sqrt{x/a}+\sqrt{y/a}=1 $$
in the earlier "rectangular parabola"
is recognized as representing a parabola with axis of symmetry inclined at $\pi/4$ to x-axis, touches the coordinate axes tangent at $x=a,y=a,$ it is not difficult to see that 
$$ \sqrt{x/a}+\sqrt{y/b}=1 $$
represents a parabola with axis inclined at $\tan^{-1}(b/a)$ to x-axis touching the axes at $x=a,y=b.$
I had included a sketch of this particular situation in the above link.
