Level Curves of xy/(x^2+y^2)

• Sep 24th 2012, 11:10 PM
dnftp
Level Curves of xy/(x^2+y^2)
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.
• Sep 25th 2012, 01:25 AM
tom@ballooncalculus
Re: Level Curves of xy/(x^2+y^2)
Quote:

Originally Posted by dnftp
this is an ellipse

No because you have minus xy, not minus constants ab, i.e.

x&#94;2&#43;y&#94;2-xy&#47;k&#61;0 - Wolfram|Alpha

not

ellipse - Wolfram|Alpha

E.g. for k = 1/4

x&#94;2&#43;y&#94;2-4xy&#61;0 - Wolfram|Alpha

and k = 1/2

x&#94;2&#43;y&#94;2-2xy&#61;0 - Wolfram|Alpha

and k from -1/2 to 1/2

z&#61; &#40;xy&#41;&#47;&#40;x&#94;2 &#43; y&#94;2&#41; - Wolfram|Alpha
• Sep 25th 2012, 07:17 AM
Someone2841
Re: Level Curves of xy/(x^2+y^2)
Quote:

Originally Posted by dnftp
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.

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 = $\frac{x/k \text{ + or - } \sqrt{x^2/k^2 - 4x^2}}{2}$ (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.