<s>[INST] The parametric ratio $\frac{x}{y}$ with known $x+y$ and $x\cdot y$ $x$ and $y$ are in fact $\lambda_1$ and $\lambda_2$, the bigger and smaller eigenvalues of a parametric matrix $A'A$, and $t$ is a very small constant. I have that
$$
\begin{split}
x+y &= 1+ \frac{t^2}{2} \\
 xy &= \frac{t^2}{4}
\end{split}
$$
How would I prove that $\frac{x}{y}\ge \frac{1}{t}$? The solutions of the system are here, but it gets pretty messy. Is there an elegant way to go about this?
 [/INST] So $x=\frac{1}{4}(t^2+2+\sqrt{t^4+4})$ and $y=\frac{1}{4}(t^2+2-\sqrt{t^4+4})$,  $\dfrac{x}{y}=\dfrac{(t^2+2+\sqrt{t^4+4})^2}{4t^2}$ say $t>0$. 
To prove that  $$\dfrac{(t^2+2+\sqrt{t^4+4})^2}{4t}\ge 1\Leftrightarrow t^4+4+4t^2+t^4+4+2(t^2+2)(\sqrt{t^4+4})-4t\ge 0$$ but $t^2-t+1\ge 0$.
</s>