I'm trying to sketch level curves for $f(x,y)=x/(x^2+y^2)$ at $c=-2,0,4$. If $c=0$ we know that $C\_0=\{(x,y)\in \mathbb{R}^2 \setminus (0,0):x=0\}$. The other cases are a little tougher.

For $c=-2$ we can solve

$$\frac{x}{x^2+y^2}=-2 \iff y= \pm \sqrt{-x^2-x/2}, x\in [-1/2,0]$$

But I don't know how to sketch that. Alternatively, in polar

$$\cos \theta= -2r$$

But how do I know which are the bounds on $\theta,r$? Also, how do I sketch that curve? I'm really lost.