4. A circle in the plane divides the plane into two regions: the region inside t

Question

4. A circle in the plane divides the plane into two regions: the region inside the circle and the region outside the circle. If two circles intersect in two points they divide the plane into four regions: (1) outside both circles, (2) inside the intersection of the two circles, (3) the interior of the first circle that is not in the second, and (4) the interior of the second that is not in the first. A straight line that intersects a circle in two points divides the plane into four regions: two inside the circle and two outside of the circle. What is the maximum number of regions formed by 3 circles and 2 straight lines? Find a formula for the maximum number of regions formed by m circles and n straight lines. Explain how you derived your formula.

Explanation / Answer

Each pair of line can at the max intersect at one point. Therefore if you have n lines there can be nC2 pairs each with 1 intersection. So nC2 intersections.

Similarly for each pair of circle can at the max intersect at 2 points. Therefore if you have m circles there can be mC2 pairs of circles each with 2 intersections. So the total number of intersections will be 2*mC2.  

Also, a line and a circle can intersect at 2 points (at the max). Therefore if there are n lines and m circles there can be n*m combinations of lines and circles each with 2 intersections. Hence the total number of intersections will be 2*m*n.

Therefore the general formula for maximum number of intersections of n lines and m circles is

nC2 + 2*(mC2) + 2*m*n

m=3 and n=2

no of intersections = 2C2 + 2(3C2) + 2*3*2 = 19 => number regions formed is twice of intersection points = 2*19 = 38 regions.

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