This is exercise 18. d) from Spivak Calculus. Some things I have proven from previous problems that I was using to help me figure this out are:
$x^2+y^2\geq 2xy$; equality when x and y are both 0
I believe I know the answer however I don't think my method is leading to a valid proof.
Here is my process, I rearranged:
I then considered what value -a, must be in order for this inequality to not hold:
So I want to find where $(x^2+y^2)/xy<-a$
$x^2+y^2\geq2xy$ $\Rightarrow (x^2+y^2)/xy\geq2$
Therefore if $(x^2+y^2)/xy<-a$ then $-a>2$. It follows that if xy<0 that -a<-2.
This is about where I'm stuck. I know I can't simply say that since $(x^2+y^2)/xy>-a$ is not true when $-a>2$ or $-a<-2$. That I then know the interval where it is true is $-2<-a<2$.
There's some other odd things I thought about but I'm not sure if they go anywhere. I've spent some time thinking about how x^2+y^2 changes compared to $axy$, how many counterexamples there are when a>2. I also had proven $x^2+xy+y^2>0$ as the sum of 2 squares previously but I don't think it helps me here.