Write the equations of the rectilinear generatrices of the hyperbolic paraboloid $\frac{x^2}{p}-\frac{y^2}{q}=2z\ \ ,p,q>0$. Out of these rectilinear generatrices select those which are parallel to the plane $\frac{x}{\sqrt{p}}-\frac{y}{\sqrt{q}}=0$, show that they are pairwise noncoplanar and compute the distance between any two of them.
I know I have to do some system of equations like: $$\frac{x}{\sqrt{p}}=\lambda \frac{y}{\sqrt{q}} \ \ \ , z=\mu \ \ \ , \frac{x^2}{p}-\frac{y^2}{q}=2z$$
But this won't result in an equation only with $\lambda$ and $\mu$. Cand somebody help me, please?
Do you agree that, by taking the product of the two equations, we have the implication $$\underbrace{\begin{cases}(E_1) \ &\frac{x}{\sqrt{p}}-\frac{y}{\sqrt{q}}&=&\lambda\\ (E_2) \ &\frac{x}{\sqrt{p}}+\frac{y}{\sqrt{q}}&=&\frac{1}{\lambda} 2z\end{cases}}_{\text{line} \ L_{\lambda}} \ \ \implies \ \ \underbrace{\frac{x^2}{p}-\frac{y^2}{q}=2z}_{\text{Hyp. paraboloid}}$$
You know that implication of properties is inclusion of the corresponding sets, i.e., the inclusion of a line (indexed by a parameter, therefore an infinite number of lines) into the hyp. paraboloid.
Why do I say we have a line ? Because it is the intersection of two planes (there is an implicit "and" between the two equations).
This family of lines is parallel to the plane with equation
$$\text{plane (P) :} \ \ \ \frac{x}{\sqrt{p}}-\frac{y}{\sqrt{q}}=0\tag{1}$$
because, apart the case $\lambda=0$, it is impossible to have simultaneously $(E_1)$ and (1) ; therefore no common point can exist between line $(L_{\lambda}) and plane (P).
Caution : you must know that there is another family of lines, different from the first one:
$$\underbrace{\begin{cases}\frac{x}{\sqrt{p}}-\frac{y}{\sqrt{q}}&=&2 \mu z\\ \frac{x}{\sqrt{p}}+\frac{y}{\sqrt{q}}&=&\frac{1}{\mu} \end{cases}}_{\text{line} \ L'_{\mu}} \ \ \implies \ \ \underbrace{\frac{x^2}{p}-\frac{y^2}{q}=2z}_{\text{Hyp. paraboloid}}$$