Question

Draw different pairs of circles. How many points does each pair have in common? What is the maximum number of common points?

Answer 1

Step 1. Draw two circles when they do not intersect with each other.

From the above figure, it is observed that there are no common points.

Step 2. Draw two circles when they intersect with each other at single point.

From the above figure, it is observed that there is only one common point $Y$.

Step 2. Draw two circles when they intersect with each other at two points.

From the above figure, it is observed that there are two common points $P$ and $Q$.

There is no other possibility of two circles intersecting each other.

Therefore, two circles have at the most two points in common.

Hence, the maximum number of common points is $2$.