I'm working through a multivariable calculus course. I'm ~5 years removed from my most recent calculus course, and ~10 years removed from my most recent trigonometry course. While I'm understanding the new material pretty easily, I keep tripping up on material that I'm (understandably) expected to already know.
One review exercise is:
Draw the level curve for $f(x, y) = \frac{x}{x^2 + y^2}$ at values $c = -2, 0, 4$.
For $c = 0$, this is trivial ($x = 0, y \ne 0$). I'm having a bit more trouble with the other values. I can look in the back of the book and see that the shapes are two circles. But what signs am I looking for to tell me this?
I can simplify the equation of the level curve to something like this:
$$c = \frac{x}{x^2 + y^2}$$ $$c(x^2 + y^2) = x$$ $$cx^2 - x + cy^2 = 0$$ $$cy^2 = -cx^2 + x$$ $$y^2 = -x^2 + \frac{x}{c}$$ $$y = \pm \sqrt{-x^2 + \frac{x}{c}}$$
For $x \ne 0, c \ne 0$.
I also recall that equations like this describe a ellipse:
$$a(x - x_0)^2 + b(y - y_0)^2 = r^2$$
I assume there is a similar form that accounts for the possibility of an $x$ term. But I'm not sure how to go about determining this form.
This problem isn't assigned as homework, but I'm marking it as homework anyway, because it relates to a class that I'm taking.