Homework: Problem concerning first fundamental form Here's a strange problem in our differential geometry textbook.

At a point on surface $\mathbf{r}=\mathbf{r}(u,v)$, the equation $Pdudu+2Qdudv+Rdvdv=0$ determines two tangential directions. Prove that these two tangential directions are normal iff $$ER-2FQ+GP=0$$
  where $E=\langle\mathbf{r}_u,\mathbf{r}_u\rangle,F=\langle\mathbf{r}_u,\mathbf{r}_v\rangle,G=\langle\mathbf{r}_v,\mathbf{r}_v\rangle$

I don't understand what this problem wants me to do. Since the statement :"the equation $Pdudu+2Qdudv+Rdvdv=0$ determines two tangential directions" is a bit ambiguous. Hope to find some good understanding of this problem, no need for solutions, thanks!

Well we know the first fundamental form gives: $$\begin{align} & {} \quad \mathrm{I}(aX_u+bX_v,cX_u+dX_v) \\ & = ac \langle X_u,X_u \rangle + (ad+bc) \langle X_u,X_v \rangle + bd \langle X_v,X_v \rangle \\ & = Eac + F(ad+bc) + Gbd. \end{align}$$ 
And we have the unit normal vector to a tangent plane at a point as: $$n = \frac{{\bf x}_u\times {\bf x}_v}{ |{\bf x}_u\times {\bf x}_v|}.$$
So if you combine your two equations $ER-2FQ+GP=0$ and $Pdudu+2Qdudv+Rdvdv=0$ and replace your $E,F,G$ by there respective inner products then I think you will get your desired outcome? I hope I am going in the right direction. 
I will delete this answer if it is wrong. Hopefully someone can answer this question, seems interesting to me. 
