Draw level curves for $f(x,y)=\frac{x^2+y^2}{xy}$

Question

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

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

$z=0\to x^2+y^2 = 0$, which is a circle of radius $0$, or nothing.

$z=1\to x^2+y^2=xy\to \color{red}{x^2-xy+y^2=0}$

How do I even graph this?

How do I know what it looks like.

I tried plotting on Wolfram Alpha but the result is literally nothing.

Answer 1

Update....

convert to polar coordinates.

$f(r,\theta) = \frac {1}{\sin\theta\cos\theta} = 2\csc 2\theta$

Each contour is a ray with an open end at the origin.

and $f(x,y)$ is undefined at $(0,0)$

Answer 2

Take the two values such that $a+b=z$ and $ab=1$ so you have for each value of $z$ the two right lines $$y^2+x^2-zxy=(y-ax)(y-bx)=0$$

It is clear that for each value of your $z$ $a$ and $b$ are univocally determined by an equation of second degree and $a$ and $b$ can be real or non-real.

Example.- $z=4$ gives the two lines $$(y-0.268x)(y-3.732x)=0$$ where the numerical values are approximates (What are the exact values in this example?)

Answer 3

For some reason this would not work in a comment. WA worked for me, with this link