This problem appears in Shafarevich's Basic Algebraic Geometry.
Let $Q\in\mathbb{P}^3$ be an irreducible quadric surface and $\Lambda_X\in\Pi$ he set of points on the Plucker hypersurface $\Pi\in\mathbb{P}^5$ corresponding to lines contained in $Q$. Prove that $\Lambda_X$ consists of two conics.
Associating $Q$ with a symmetric billinear form $M$. Lines in $Q$ correspond to pairs of points $a, b$ in $\mathbb{C}^4$ satisfying $(at+b)^\intercal M(at+b)=0$, which is equivalent to $a^\intercal Ma=a^\intercal Mb=b^\intercal Mb=0$. However, I don't know how to translate this into the coordinates of $\Pi$. These coordinates should be the six coordinates of the form $x_i\wedge x_j$. How should I proceed? Do I just square and multiply these terms and hope something pops out?
Instead of treating the general case, I show here the concrete example of the two families of lines drawn on a particular quadric, the hyperbolic paraboloid (see an architectural point of view on this surface on the first pages of this document : https://www.bebee.com/producer/@lada-prkic/magnificent-hyperbolic-paraboloid) with the simple equation
$$(HP) \ \ \ z=xy \tag{1}$$
These families of lines can be given the following representations :
$$(L_a)\begin{cases}a=y\\z=ax\end{cases} \ \ a \ \text{being an arbitrary real number} \ \neq 0\tag{2}$$
for the first family (multiplying together these equations implies (1) which means that lines $L_a$ are included into (HP)).
The second family has equations :
$$(\Lambda_b)\begin{cases}b=x\\z=by\end{cases} \ \ b \ \text{being an arbitrary real number} \ \neq 0. \tag{3}$$
Now, turn to a point-direction representation of the two families which are resp.
$$(L_a) \ \ \begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}x\\a\\ax\end{pmatrix}=\begin{pmatrix}0\\a\\0\end{pmatrix}+x\begin{pmatrix}1\\0\\a\end{pmatrix}$$
and
$$(\Lambda_b) \ \begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}b\\y\\by\end{pmatrix}=\begin{pmatrix}b\\0\\0\end{pmatrix}+y\begin{pmatrix}0\\1\\b\end{pmatrix}$$
whose Plücker coordinates (https://en.wikipedia.org/wiki/Pl%C3%BCcker_coordinates)
$$(\underbrace{d_1,d_2,d_3}_d;\underbrace{m_1,m_2,m_3}_m)$$
are
$$L_a : (0,a,0;a^2,0,a) \ \ \ \text{and} \ \ \ \Lambda_b: (b,0,0;0,-b^2,b),$$
which are second degree curves situated on Plücker surface with equation : $$d.m=d_1.m_1+d_2.m_2+d_3.m_3=0$$