If it is a line, rather than a line segment, then the circle intersects the line if the distance from the centre of the circle to the closest point on the line is smaller than the circle's radius.
If you imagine moving a circle around near the line, and whenever the circle intersects the line, you draw a red line from the middle of the circle to the line, you'll find the closest point on the line to the circle always makes a right angle to the line towards the circle's centre.
Now if you have a line segment, the way to think about it is in three sections - the line segment; the area beyond one end of the line segment; and the area to the other end of the line segment. If your circle's centre is in the area defined by the line segment swept along a direction perpendicular to the line segment, then you can use the above calculation to find the distance from the centre of the circle to the line. Otherwise, it becomes a simple matter of finding the distance from the centre of the circle to whichever end of the line segment is closer.
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