Is a positive integer) from both the origin and the point (a,b) is equal to the absolute difference between a and b.

To see why this is true, consider a point on the X axis that is c units away from both the origin and (a,b). Let this point have coordinates (x,0). Then we have:

- Distance from (x,0) to the origin = x
- Distance from (x,0) to (a,b) = sqrt((x-a)^2 + b^2)

Since we want these distances to be equal to c, we can set up the following equations:

x^2 = c^2
(x-a)^2 + b^2 = c^2

Solving for x in the first equation, we get x = c or x = -c. Substituting these values into the second equation and simplifying, we get:

c = abs(a-b) or c = abs(a+b)

(Note that we took the square root of both sides of the second equation, which introduces a plus/minus sign. However, we can ignore the negative solution since we are only interested in points on the positive X axis.)

Therefore, if c = abs(a-b), then there is exactly one point on the X axis that satisfies both distance conditions (namely, the point (c,0)). If c = abs(a+b), then there are exactly two points on the X axis that satisfy both distance conditions (namely, the points (-c,0) and (c,0)).

In either case, the number of points is equal to the absolute difference between a and b. For example, if a = 5 and b = 2, then c = abs(5-2) = 3, and there is exactly one point on the X axis that is 3 units away from both (0,0) and (5,2): namely, the point (3,0). Similarly, if a = 5 and b = 8, then c = abs(5+8) = 13, and there are exactly two points on the X axis that are 13 units away from both (0,0) and (5,8): namely, the points (-13,0) and (13,0).

0,1,2,3