Take a line $L$ of the first type on a smooth cubic threefold $X$ over $\mathbb C$, then its normal bundle $N_{L|X}$ is isomorphic to $\mathcal{O}_L\oplus \mathcal{O}_L$. This is equivalent to say that there is a $\mathbb P^1$-family of quadric surfaces in $\mathbb P^4$ tangent to $X$ along $L$. I'm trying to write down these quadric surfaces explicitly.

Let $L=\{x_2=x_3=x_4=0\}$, then up to change of coordinates, $X$ has equation

$$x_2x_0^2+x_3x_0x_1+x_4x_1^2+\text{higher order terms in }x_2,x_3,x_4.$$

The dual map at $p={(x_0,x_1)}\in L$ is $\mathcal{D}(p)=[0,0,x_0^2,x_0x_1,x_1^2]$, which determines the hyperplane $T_{p}X$ at $p$ and we just need to find quadric surfaces containing $L$ and have tangent planes at each $p\in L$ contained in $T_{p}X$.

I can find two of such quadric surfaces:

$$x_4=0,~x_2x_0+x_3x_1=0,$$ $$x_2=0,~x_4x_1+x_3x_0=0.$$

Unfortunately, the family is not a linear combination of them and I cannot find any more such quadric surface. Note that in the 1972 paper The intermediate Jacobian of the cubic threefolds by Clemens and Griffiths, page 309, some construction are given in terms of equations of varieties of lines of the quadric surfaces in the Grassmannian $Gr(2,4)$. However, there seems to be a typo in the defining equations (of curve $B(\alpha_0,\alpha_1)$ in the paper), which I couldn't fix.

How to find the entire $\mathbb P^1$-family (hopefully in equations)? Any comments or suggestions will be appreciated!