I want to find the general equation of the two lines of intersection of a...

Step 1
The 2 families of skew lines $L_a$ and $L_b^{\prime }$ generating hyperboloid with one sheet (H) can be retrieved, starting from its equation
$\begin{array}{}\text{(1)}& x^2+y^2-z^2=1⟺(y-z)(y+z)=(1-x)(1+x),\end{array}$
in the following natural way:
Lines $\begin{array}{}\text{(2)}& \text{Lines}{L}_a:\{\begin{array}{lll}y-z& =& a(1-x)\\ y+z& =& {\displaystyle \frac{1}{a}}(1+x)\end{array}\end{array}$
Lines $\begin{array}{}\text{(3)}& \text{Lines}{L}_b^{\prime }:\{\begin{array}{lll}y-z& =& b(1+x)\\ y+z& =& {\displaystyle \frac{1}{b}}(1-x)\end{array}\end{array}$
for any non-zero real number a or b.
Indeed: by multiplication of its 2 equations, (2) $⟹$ (1) ; implication of equations meaning inclusion of corresponding geometric entities ( $\mathrm{\forall }a,L_a\subset H$ ) as desired. For the same reason, $\mathrm{\forall }b,L_b^{\prime }\subset H$ .
Therefore, for a given point $(x_0,y_0,z_0)$ , you just have to find the values of coefficients a and b, which is straightforward.
Consider the case of a. From the first equation in (2), one gets:
$\begin{array}{}\text{(4)}& a=\frac{y_0-z_0}{1-x_0}=\frac{y_0\pm \sqrt{x_0^2+y_0^2-1}}{1-x_0}\end{array}$
which is valid under the condition that $x_0\ne 1$ . If $x_0=1$ , get a instead from the second equation in (2).
Do the same for b and plug these expressions into (2), resp. (3).