How to find an equation of all planes that pass through points $(a,0,a^2)$ and $(a^{-1},0,a^{-2})$? I know that there are infinity of them, but can we write it down in one equation, possibly using some variables?
I found two planes that pass through these points: one is x-z plane and it is given by equation $y=0$, and second plane passes through these points and also perpendicular to x-z plane and it is given by equation $z=(a+a^{-1})x-1$.
Best Answer
Actually, you already did most of the job.
Multiply the equation of the first plane by $\sin\varphi$, the second one by $\cos\varphi$, and add both together. That would be your single equation with a parameter ($\varphi$).
$$y\sin\varphi + z\cos\varphi - x\left(a+\frac1a\right)\cos\varphi+\cos\varphi = 0$$
How so?
Well, both equations are certainly true at both points, hence so is any linear combination of them. Our resulting plane passes through both points as well.
How do I know that the equation covers all such planes?
Why, just find the angle between my plane and your second plane, and realize that it is $\varphi$. By varying $\varphi$, you rotate my plane around the line that connects the two points.
So it goes.