Prove that the line $\textbf{l}$ tangent to a conic $C$ at a point $\textbf{x}$ on C is given by $\textbf{l}=C\textbf{x}$. The equation of a conic in matrix form is defined as $\textbf{x}^TC\textbf{x}=0$.
The proof in the textbook says:
The line $\textbf{l}=C\textbf{x}$ passes through $\textbf{x}$, since $\textbf{l}^T\textbf{x}=\textbf{x}^TC\textbf{x}=0$. If $\textbf{l}$ has a one-point contact with the conic, then it is a tangent, and we are done. Otherwise suppose that $\textbf{l}$ meets the conic in another point $\textbf{y}$. Then $\textbf{y}^TC\textbf{y}=0$ and $\textbf{x}^TC\textbf{y}=\textbf{l}^T\textbf{y}=0$. From this it follows that $(\textbf{x}+\alpha \textbf{y})^TC(\textbf{x}+\alpha \textbf{y})=0$ for all $\alpha$, which means that the whole line $\textbf{l}=C\textbf{x}$ joining $\textbf{x}$ and $\textbf{y}$ lies on the conic C, which is therefore degenerate.
What I am confused about is where this:
From this it follows that $(\textbf{x}+\alpha \textbf{y})^TC(\textbf{x}+\alpha \textbf{y})=0$ for all $\alpha$
comes from, why is this statement true? I don't get how the preceding sentence implies this.
We simply have $$(x+\alpha y)^TC(x+\alpha y)=x^TCx+\alpha x^TCy +\alpha y^TCx+\alpha^2y^TCy$$ and each term on the right side is $0$, using $y^TCx=x^TCy$.