Let $F(x,y)=ax^2+2bxy+cy^2+2dx+2ey+f,$ $\phi(x,y)=ax^2+2bxy+cy^2,$ $x,y \in \mathbb{R}$. Assume that for some $x_0, y_0 \in \mathbb{R}$ and for some $\alpha, \beta \in \mathbb{R}$ such that $\alpha^2+\beta^2>0$ following conditions hold: $\phi(\alpha,\beta)=0,$ $\alpha(ax_0+by_0+d)+\beta(bx_0+cy_0+e)=0,$ $F(x_0,y_0)=0.$
Is it then $ W:=\left | \begin{array}{rrr} a & b & d\\ b & c & e\\ d & e & f \end{array} \right |=0 \ \ \ ? $
It concerns the following question from analitical geometry. Consider a curve $F(x,y)=0$. Let a strightline on a plane has equations $x=x_0+\alpha t,$ $y=y_0+\beta t,$ where $[\alpha,\beta]$ has asymptotic direction, i.e. $\phi(\alpha,\beta)=0.$ Assume that this line lies on the curve, i.e. equation $\phi(\alpha,\beta)t^2+2[(\alpha(ax_0+by_0+d)+\beta(bx_0+cy_0+e)]t+F(x_0,y_0)=0$ is satisfied by every $t \in \mathbb{R}$. Then the curve $F(x,y)=0$ has to be degenerated.
It is clear if we used theorem about classification of curves of the second order. But I look for purely algebraic explanation.
Thanks.