Points on the arc and the distance between them

Question

Let $r(t) = (2t,t-1,2-2t)$ be an arc. $A= (-2,-2,4)$ is a point on it. Which two points on an arc are on distance 6 from A?

Answer 1

First, the "arc" is actually a line in space and point $A$ indeed is on it ($t=-1$). What we can do is using the distance formula. We can set up the distance between any arbitrary point $(2t,t-1,2-2t)$ and $(-2,-2,4)$ and equate this distance to $6$. Assuming you know the distance formula: $\sqrt{(-2-2t)^2+(-2-t+1)^2+(4-2+2t)^2}=6$. Simplifying and squaring both sides: $9t^2+18t+9=36$. This is a quadratic equation that is perfectly factorable (gives you two nice $t$ values), but can you do that? PS: Does it make sense that you get two $t$ values?

Answer 2

HINT:

We want the distance to be $$\sqrt{(2t+2)^2+(t-1+2)^2+(2-2t-4)^2}=6$$ Can you now collect the like terms and solve the quadratic?