find the coordinates of the points which are simultaneously 6 units from the points (6,6) and equidistant from the x and y axis. There are two ways to do this. I would like them both.

Answer 1

equidistant from the x,y axes means that if (x,y) is one point, y=x or y=-x

on the line y=x, if the distance from (6,6) is 6, then the x- and y-distances are ±6/√2.

So, (6-6/√2,6-6/√2) and (6+6/√2,6+6/√2) are two such points.

Points on the line y = -x are too far away from (6,6) to qualify.

Answer 2

I guess you could also do this by finding the intersection of

(x-6)^2 + (y-6)^2 = 36
and
y=x

the answers are the same.