How to visualize the partition created by this relation? (Please see image)

Question

I don't understand how to answer part (c) of this question, to which the answer is: concentric circles centred at the origin.

I know $$x^2+y^2=r^2$$ is the equation of a circle, but I'm having trouble understanding how the concentric circles are formed.

Thanks for the help.

Answer 1

As you mentioned the equivalence classes are circles.

For example the class of $(3,4)$ is a circle of radius $5$ centered at the origin.

Now the partition ste is the set of all circles centered at the origin.

Note that these circles are disjoint and the union of all these circles is the entire $(x,y)$ plane.

Thus the (x,y) plane is partitioned into these disjoint circles.