Knowing the distance from origin of two points, can you infer their distance from each other? This would be in a singular quadrant, so the coordinates of both points would be positive. I am wondering if, given two points and both of their distances from the origin, could you evaluate their distance from each other?
Say I have Point A, and Point B.
I know Point A has positive coordinates, and is X distance away from the origin.
Point B also has positive coordinates, and is distance Y from the origin.
I don't want to actually calculate the Euclidean distance between the points, but rather infer if Point A and Point B are within some variable Epsilon of each other. Given only this information, is it possible to find a solution?
 A: Hint
Geometrically, $A$ can be located anywhere in the first quadrant of a circle with center at zero and radius $X$, and $B$ - on a quarter circle centered at the origin with radius $Y$.
You need to find places where these quarter circles are closest and farthest from each other. Does this help?

Speculation
Didn't prove this conclusively but it seems very geometrically motivated that the max distance is achieved when one lies on the $x$-axis and the other on the $y$-axis, which would cause the max distance to be from $(0,X)$ to $(Y,0)$, a total of $\sqrt{Y^2+X^2}$.
A: Alternative perspective:
The Law of Cosines is controlling here.
Assume that points A and B are in $\Bbb{R^2}$, let O denote the origin, whose coordinates are $(0,0)$, and consider $\triangle$ ABO.
Let $a,b,c$ denote the lengths of the respective line segments $~\overline{AO}, ~\overline{BO}, ~$ and $~\overline{AB}.$
Let $\theta$ denote $\angle$ AOB.
Then:

*

*$a,b~$ are known.

*$c~$ and $~\theta~$ are unknown.

*$c^2 = a^2 + b^2 - 2ab[\cos(\theta)].$
Therefore, $~c~$ can not be determined unless $~(\theta)~$ is specified.
A: To picturize, consider the following image:

Here, If you are given that point $A$ is at a distance of $2$ units from the origin and point $B$ is at a distance of $4$ units from the origin, then from the figure, the point $A$ may lie anywhere on the arc $BC$, which is a part of a circle with radius $2$ units from the origin while the point $B$ may lie anywhere on the arc $DE$ (part of circle of radius $4$ units).
Assume point $A$ is at $F$ then if point $B$ is at $G$ then distance between $A$ and $B$ is $2$ units. If $B$ were at $H$ or $E$ then the distance would have been $2.32$ units and $2.96$ units respectively.
Thus, as pointed out by user2661923, unless we don't know the angle made by the line segment joining the $2$ points, we can't determine the distance between the $2$ points.
