Question

Find two level curve of f(x,y)=2xy/(x^2+y^2 ) . Give a rough sketch

Answer 1

$$f(x, y)=\dfrac{2xy}{x^2+y^2}$$

$$z=\dfrac{2xy}{x^2+y^2}$$

$$c=0: \dfrac{2xy}{x^2+y^2}=0$$

$$x=0, y\not=0$$

Or

$$y=0, x\not=0$$

$$c=1: \dfrac{2xy}{x^2+y^2}=1$$

$$x^2-2xy+y^2=0$$

$$(x-y)^2=0$$

$$x=y, x\not=0$$

Blue graph

$$c=-1: \dfrac{2xy}{x^2+y^2}=-1$$

$$x^2+2xy+y^2=0$$

$$(x+y)^2=0$$

$$y=-x, x\not=0$$

Violet graph

$$c=0.5: \dfrac{2xy}{x^2+y^2}=0.5$$

$$y^2-4xy+x^2=0$$

$$y=\dfrac{4x\pm\sqrt{16x^2-4x^2}}{2}$$ $$y=(2-\sqrt{3})x\ or\ y=(2+\sqrt{3})x, x\not=0$$

Red graph