I am trying to find the level curves for the function g(x,y)= k = xy/(x^2+y^2).I get, x^2+y^2-xy/k=0. I know this is an ellipse, but I do not know how to factor, and find values of k for which the level curves exist.
No because you have minus xy, not minus constants ab, i.e.
x^2+y^2-xy/k=0 - Wolfram|Alpha
not
ellipse - Wolfram|Alpha
E.g. for k = 1/4
x^2+y^2-4xy=0 - Wolfram|Alpha
and k = 1/2
x^2+y^2-2xy=0 - Wolfram|Alpha
and k from -1/2 to 1/2
z= (xy)/(x^2 + y^2) - Wolfram|Alpha
I think the quadratic equation works to solve for y in terms of x and k.
y^2 - x/k y + x^2 = 0 -> y = $\displaystyle \frac{x/k \text{ + or - } \sqrt{x^2/k^2 - 4x^2}}{2}$ (didn't know how to make the + or - symbol in tex)
Check this as I did it in a hurry! I think the logic is sound, though.