There is a trick to accurate circles

If you want accurate circles, then do not draw them with by dragging the ellipse tool. It is not a circle tool (you can get one here). Using ellipse tool like this would be way too inaccurate. Instead what you do is you draw a line that represents your radius. Then measure your line and alt click on the point you want the center to be at and TYPE in the radius.

Image 1: When you alt click you get a dialog to type the accurate radius you measured with the line tool

This is way more accurate, and it is even more accurate in illustrator CC (since you can get more digits on the line)

Image 2: If you want accurate circle intersections TYPE the radius from meaurement

If you want to find intersections between other things, same thing applies except that you need to use smart guides on your circle.

Since this can get boring to do over time Ive written a script that will draw a circle if you draw the radius line (since i can not always install subscribe):

var sel = app.activeDocument.selection;
for(i=0; i<sel.length;i++){
   var pts = sel[i].pathPoints;
   var radius = Math.sqrt(
          Math.pow(pts[1].anchor[0]-pts[0].anchor[0], 2)+
          Math.pow(pts[1].anchor[1]-pts[0].anchor[1], 2)
          );
          
    app.activeDocument.layers[0].pathItems.ellipse( 
        pts[0].anchor[1]+radius, 
        pts[0].anchor[0]-radius, 
        radius*2, radius*2);
}