Find the distance between two sets in $\mathbb E^2$

Question

Find the distance between two sets in $\mathbb E^2$:

P = {$(x,y): x+2y = 4$} and K = {$(x,y): $$(x+1)^2 + (y+1)^2 = 1$}

Need some help with this one

Answer 1

We can make a change of variables and replace $x$ by $x-1$ and $y$ by $y-1$.

$$x \mapsto x-1$$ $$y \mapsto y-1$$

Now the problem becomes find the distance between the line $x+2y=7$ and the circle $x^2 + y^2 =1.$

The line through the origin and perpendicular to this line intersects the line and the circle at the closest points. This line is $2x-y=0.$

Solving the system of two lines yields $x=7/5$ and $y=14/5.$

This point is a distance $7/\sqrt{5}$ from the origin and $7/\sqrt{5}-1$ from the circle.

Distance between the sets is $7/\sqrt{5}-1$.

Answer 2

The hint:

It's just $$\frac{|1\cdot(-1)+2\cdot(-1)-4|}{\sqrt{1^2+2^2}}-1.$$