Minimum distance from a point to a line

Statement

Find the point on the line $y=2x+1$ closest to the origin and the corresponding minimum distance.

Solution

A point on the line is $(x,\ 2x+1)$. Minimise the squared distance from the origin: $D^2(x)=x^2+(2x+1)^2=5x^2+4x+1.$ $\dfrac{d}{dx}D^2=10x+4=0\Rightarrow x=-0{.}4$, with $y=2(-0{.}4)+1=0{.}2$.

Distance: $d=\sqrt{(-0{.}4)^2+(0{.}2)^2}=\sqrt{0{.}2}\approx 0{.}447$.

$\boxed{P(-0{.}4,\ 0{.}2),\ d_{\min}=\sqrt{0{.}2}\approx 0{.}447}$