Sketch the open ball of a metric

Question

In $\mathbb{R}^2$ sketch B((1,2),3), the open ball of radius 3 at the point (1,2) with the following metric....

$d(x,y)= \dfrac{5||x-y||_2}{1 + ||x-y||_2} $

I know what the sketch looks like but I don't know how to compute it. Please help

Answer 1

First of all, note that $d(x,y)<5$ for all $x,y\in\Bbb R^2.$ (Why?)

Second, note that for any $0\le k<5$, the following are equivalent (why?): $$d(x,y)<k\\5\lVert x-y\rVert_2<k+k\lVert x-y\rVert_2\\(5-k)\lVert x-y\rVert_2<k\\\lVert x-y\rVert_2<\frac{k}{5-k}$$