How can we draw the contour line?

Question

Let $f:\mathbb{R}_+^2\rightarrow \mathbb{R}$ with $y=f(x_1,x_2)=\sqrt{x_1 \cdot x_2}$.

I want to draw the contour line $y=2$.

$$$$

We have that $\sqrt{x_1\cdot x_2}=2\Rightarrow x_1\cdot x_2=4$.

How can we draw the contour line? What kind of curve is this?

$$$$

EDIT:

For $x_1=1$ we get $x_2=4$, for $x_1=2$ we get $x_2=2$, for $x_1=4$ we get $x_2=1$ so do we get :

?

Answer 1

If $x_1x_2=4$ and $x_i \neq 0$, then $$x_2 = \frac{4}{x_1}$$ This a hyperbola from high school, think $y=\frac{1}{x}$ in good old-fashioned $xy$-coordinates.