This is one homework problem and hence I want some hint but not a whole answer.
Let $P$ be a projective space and $X\subset P$ be a non-singular variety. Prove that the collection $L_p$ of lines contained in $X$ that pass through $p$ defines a closed subset of the projectivized tangent space $\mathbb{P}(T_p X)$.
In other word, we want to show that such defined $L_p$ defines a closed subset of $\mathbb{P}(T_p X)$:$$L_p = \{l:p\in l\subset X, l\text{ is a 1-dimensional linear subspace in P}\}.$$
I cannot see the connection between $L_p$ and $\mathbb{P}(T_p X)$. Please try to explain the connection using elementary knowledge.
Thanks.
Do you see how a line through $P$, lying in $X$, gives a tangent direction to $X$ at $P$? This is the first step, and if you don't see it right away, try drawing a picture of a surface with a line lying on it (e.g. a quadric in space), and then see if you see it.
Added: As discussed in comments (assuming that we are working with closed points of varieties over an alg. closed field $k$), if $f : X \to Y$ is a morphism and $x \in X$, we get an induced morphism $f_*: T_x X \to T_y Y.$
Now apply this with the source $X$ being a line $L$ through $p$ and lying on $X$, the target $Y$ being $X$, and $f$ being the inclusion $L \subset Y$. And, course, set $x = y = p$.
Then we get a morphism $T_p L \to T_p X.$
(a) Show that when $f$ is a closed immersion, the morphism $f_*$ is an embedding. In particular, in our case, we see that $T_pL$ embeds into $T_p X$.
(b) What is $T_pL$?
(c) Put (a) and (b) together to make substantial progress on your question.