$F(x,y)= {2xy \over x^2+y^2}$ when $(x,y)$ is non-zero. And $F(x,y) = 0$ when $(x,y)$ is zero.
I tried to solve this with beginning from when $z=1$ and I can't go on when $z=c ( > 1 ).$ ( When $z=2 , (x-y/2)^2 + 3/4 y^2 = 0$ and it doesn't make sense. ) And also I can't understand polar coordinates method well.
Please help me
To convert to polar, you just have to substitute $x=r\cos\theta$ and $y=r\sin\theta$. Your level curves become $$c = \frac{2r\cos\theta r\sin \theta}{r^2\cos^2\theta + r^2\sin\theta} = 2\cos\theta\sin\theta = \sin 2\theta.$$ Those should be familiar curves(?)