$\lambda : (x,y,x^2+\lambda y^2)$ contains straight lines For which $\lambda\in \mathbb{R}$ $$\varphi(x,y) = (x,y,x^2+\lambda y^2)$$ is a ruled surface, i.e. it can also be parametrized as
$$ \mathbb{R}^2 \ni (t,u) \mapsto g(t) + uw(t) $$
with $g$ a differentiable curve and $w$ a differentiable vector field?

If $\lambda<0$, then $\lambda=-k^2$ and $x^2+\lambda y^2=(x-ky)(x+ky)$. 
Set $u=x-ky$ and $v=x+ky$. Then $x=\frac{u+v}{2}$, $y=\frac{u-v}{2k}$ and $z=uv$. 
Then:
$$
(u,v)\rightarrow \; (\frac{u+v}{2},\frac{u-v}{2k},uv)=(\frac{u}{2},\frac{u}{2k},0)+v\,(\frac{1}{2},\frac{-1}{2k},u)
$$
