A bitangent hyperplane to a projective curve

Question

Let $X\subset \Bbb P^n$ be a smooth projective curve. A hyperplane $H\subset \Bbb P^n$ is said to be tangent to $X$ at $p$ if $\text{div}(H)(p)\geq 2$. This is equivlaent to the condition that $H$ contains the tangent line to $X$ at $p$. $H$ is said to be bitangent to $X$ if it is tangent to two or more points of $X$. Suppose $H$ is tangent to $p$ and $q$. Then how do we know that $p$ and $q$ have the same tangent line? I can only see that $H$ contains the tangent lines at $p$ and $q$, but I can't see why these two are same. (This is asserted in p.220 of Miranda's book Algebraic Curves and Riemann Surfaces.)