Let there exist real numbers $a,b,c,d$ such that $\lambda x^2+2xy+y^2=(ax+by)^2+(cx+dy)^2\ \ \forall x,y\in\mathbb{R}$, then what is the condition on $\lambda$?
I think such a $\lambda$ does not exist. I got the equations $$\begin{aligned} ab+cd&=1\\ a^2+c^2&=\lambda \\ b^2+d^2&=1\end{aligned}$$ but am unable to proceed further. Any hints? Thanks beforehand.
Hint the right-hand side is never negative