Find the point on the closed segment between (0, 1) and (2, 0) that is closest to (1001, 1995)

Question

Find the point on the closed segment between (0, 1) and (2, 0) that is closest to (1001, 1995)

I do not understand what the question is asking or how to approach it

Answer 1

Let $\ell$ be a line and $P$ be a point off $\ell$. There is a point $Q\in \ell$ at a minimum distance to $P$; further, the line ${QP}$ and $\ell$ are orthogonal, i.e. their slopes are negative reciprocals.

The line connecting $(0,1)$ and $(2,0)$ is $y=1-x/2$, with slope $-1/2$. Thus the slope of the line connecting $Q$ and $P$ has slope $2$. Use point-slope form to write an equation for the line through $P$ with slope $2$, and solve: $$ 2(x-1001)+1995=1-x/2 $$Here we get $x=16/5$, which gives $y=-3/5$. But this is not within the segment, so we pick the point on this segment closest to $(16/5,-3/5)$, which is $(2,0)$.