Consider two circles on a plane given by equations :
$(x-x_{c1})^2 + (y-y_{c1})^2 = r_1^2 $
$(x-x_{c2})^2 + (y-y_{c2})^2 = r_2^2$
I need to find a pair of points $(x_1,y_1)$ on Circle 1 and $(x_2,y_2)$ on Circle two which are distance $K$ apart
By Euclidean Distance, I get
$K^2 = (x_1-x_2)^2 +[(y_{c1}-y_{c2}) + \sqrt{r_1^2 - (x_1-x_{c1})^2} - \sqrt{r_2^2 - (x_2-x_{c2})^2} ]^2 $
There are two unknowns : $x_1$ and $x_2$. I need to find one pair of such rational $x_1$ and $x_2$,
If it fits your need to just check if a pair with given distance $k$ exists or not for two circles, you can calculate the minimum distance between two points lying on two circles as well as their maximum distance. Consider different arrangements of the two circles, this will give you different minimum and maximum for the two circles.
For simple non overlapping circles, max distance would be Center Distance + radius of first circle + radius of second circle and the min distance would be Center Distance - radius of first circle - radius of second circle. Since you already have $k$ you may check if it lies between the max and min range. If it does then a point will exist for the two circles.
Some more cases are possible where the circles may overlap, are concentric or maybe have exactly same center and radius. You may calculate the max and min ranges for them yourself.