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 $\alpha^2+\beta^2>0$ the following conditions hold: $$\phi(\alpha,\beta)=0,$$ $$\alpha(ax_0+by_0+d)+\beta(bx_0+cy_0+e)=0,$$ $$F(x_0,y_0)=0.$$
Does it follow that $$ W:=\left | \begin{array}{rrr} a & b & d\\ b & c & e\\ d & e & f \end{array} \right |=0 \ \ \ ? $$
It concerns the following question from analytic geometry. Consider a curve $F(x,y)=0$. Let a straight line on a plane have 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 were to use the theorem about classification of curves of the second order. But I am looking for a purely algebraic explanation.
Thanks.