circle with common tangent, common chord

This is a plot of two circles, with color red and blue (try to drag the blue bars and red bars to change the size of the circle) They may have a common tangent or a common chord (intersect at one of two points) and the green bar controls the size of circle with the fixed intersection points. You should first try to change the size of the two circles first. See the change of it when you drag the bar. Then you may find the circle intersect at two points and drag the green bar to see the green circle.
1. Prove that the equation of the chord is just the difference between the two equations of circles. 2. What are the common points of the green circle when u drag the green bar? After all, you should know more equations of family of circles at this point, right?=]