This is kind of a subquestion of this question.
Suppose I have two distinct points $[a_0b_0:a_0b_1:a_1b_0:a_1b_1]\in Z$ and $[a_0a:a_0b:a_1a:a_1b]\in \mathbb{P}^3$ with $a_0\ne 0,\ b_0\ne 0$ where $Z=Z(X_0X_3-X_1X_2)\subset \mathbb{P}^3$. I want to understand whether it is true that for any choice of $a,b\in \mathbb{C}$ (except for $a=b=0$ obviously) the two points above lie on the same line in $\mathbb{P}^3$ (in fact, in $Z$).
For example in the case $a=b$, the second point is $[a_0:a_0:a_1:a_1]$, and this gives the unique line $$[a_0(b_0s+t):a_0(b_1s+t):a_1(b_0s+t):a_1(b_1s+t)]$$ because there is no any freedom; all quantities except for the parameters $s,t$ are determined uniquely. A similar reasoning works for $a=0, b\ne 0$ and $a\ne 0, b=0$.
But if $a\ne 0, b\ne 0$, the equation is $$[a_0(sb_0+ta):a_0(sb_1+tb):a_1(sb_0+ta):a_1(sb_1+tb)],$$ and here not only $s$ and $t$ are arbitrary, but also $a$ and $b$ are not fixed.
I'm asking this because the process of solution of the problem in the question referred to above suggests that this must determine a unique line, and I feel that I don't have enough knowledge on lines in projective spaces to say whether it's true. I'm sorry if this is too elementary.
The only thought that I have is the following: since $s,t,a$ range over $\mathbb{C}$, so does $sb_0+ta$; and similarly for $sb_1+tb$. But this doesn't mean that the line is unique, does it? Probably I should use the fact that the points are distinct, but how?