Find the image of (0,0) under the translation $x^2+y^2=1 \rightarrow x^2+y^2-2y=0$

Question

I am (slowly) working through a pure maths book as a hobby - I am retired. I have got to a section on coordinate geometry which deals with translations and change of origin

I understand a simple transformation like $x^2+y^2=4\rightarrow (x-1)^2+(y-2)^2=4$ but I cannot deal with this question, which asks:

Find the coordinates of the image of the point $(0,0)$ under a translation which maps a curve as follows:

$x^2+y^2=1\rightarrow x^2+y^2-2y=0$

I could of course just plug it into Desmos and look at the result but I don't want to do that yet. I'm trying to intuitively understand what is going on here but it isn't helping me.

Answer 1

Hint:

$$x^2+y^2-2y=x^2+(y-1)^2-1$$