I wish to draw the levels curves for:
$f(x,y)=\frac{x}{x^{2}+y^{2}}$
I have started by using the definition:
$\frac{x}{x^{2}+y^{2}}=k$
From there using algebra, and the completion to a square technique, I got:
$(\sqrt{k}x-\frac{1}{2\sqrt{k}})^{2}+ky^{2}=\frac{1}{4k}$
Which could be incorrect. I have set values for k, positive only because of the square root, and got the following level curves plot:
According to the computer, this is incorrect. The negative area of the x axis should have identical shapes to these on the positive area of the x axis. Somewhere in my calculations I got something wrong. I think I should have got
$\pm \sqrt{k}$
But not sure where and why.
In addition, the computer plots it as circles, I thought it was an ellipse. I would appreciate any assistance and clarification with this level curve.
Thank you !
i would write $$y=\pm\sqrt{\frac{x-kx^2}{k}}$$ under the condition $$\frac{x-kx^2}{k}\geq 0$$