We need to find the equation of the circle that has its
center as the midpoint between (2,-1) and (6,-5), the green circle below:

 

Midpoint between (2,-1) and (6,-5),  = (4,-3) 

and has its diameter as the distance between between (2,-1) and (6,-5)
and its radius as half that:

 

Therefore its center is (4,-3) and its
radius is one-half the diameter .

Its equation is









Or you can multiply it out and rearrange to get this:



Edwin