How to find the plane that contain certain two points?

Question

How to find the equation of the plane containing $P(2,-1,1)$ and $Q(1,0,0)$? But the answer says $ax+by+(b-a)z=a$; $a$ and $b$ not both $0$. I don't quite get the process.

Answer 1

You know the general equation of the plane in $3D$:

$$ax + by + cz = d$$

In essence, the question is asking you for non-zero $(a,b,c,d)$, up to scaling, such that the above is satisfied for $P$ and $Q$. Feeding them in,:

$$ 2a - b +c = d$$ and $$a = d$$

Must be true. This implies that the planes that contain both the points are of the form $$(a, b, b-a, a)$$

To identify them up to scaling, take the cases when $a = 0$ and $a \neq 0$ separately, to get:

$$(0, 1,1,0)$$ and $$(1, b, b-1, 1)$$

The latter for every choice of $b$. Thus, note that there are an infinite number of planes that do the job.