<s>[INST] Plot of function $(x^\frac{1}{2} + y^\frac{1}{2})^2 = 5$ Can someone explain me how can we make a plot of function:
$$ (x^\frac{1}{2} + y^\frac{1}{2})^2 = 5 $$
I tried to find a first derivative and a second derivative to understand something about this plot of function but it was useless as I understand. What should I do to plot this function by myself not by WolframAlpha. Thank you in advance.
 [/INST] $$
\begin{align}
& (x^{\frac{1}{2}}+y^{\frac{1}{2}})^{2} = 5\\
\implies & x^{\frac{1}{2}}+y^{\frac{1}{2}} = \pm \sqrt{5}\\
\implies & \sqrt{x}+\sqrt{y} = \pm \sqrt{5}
\end{align}
$$
The inequality $0\le x,y\le 5$ holds for real $x$ and $y$, as mentioned in the previous answers.
Method 1: Plot $f(x,y)=|\sqrt{x} + \sqrt{y} - \sqrt{5}|$
We take real numbers $x$ and $y$ from $0$ to $5$ and plot $f(x,y)=|\sqrt{x} + \sqrt{y} - \sqrt{5}| < \epsilon$ for very small $\epsilon$ (here $1e-9$ on a $100\times 100$ grid):

Method 2: Plot individual points
$$
x = 4y\\
\implies y = \dfrac{5}{9}\\
\implies x = \dfrac{20}{9}
$$
Similarly, $x=9y\implies (x,y)=(45/9,5/9)$, $x=16y\implies (x,y)=(80/25,5/25)$, and in general $x=n^{2}y\implies (x,y)=(\dfrac{5n^2}{(n+1)^2},\dfrac{5}{(n+1)^2})$.
Since the equation is symmetric for in $x$ and $y$, we plot both $(x,y)$ (shown in blue) and $(y,x)$ (shown in red).

Note: the point at the center of the curve is both red and blue (overlapping). Different choices of $x(y)$ give different distribution of points.
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