Draw two circles of different radii. How many points these circles can have in common? What is the maximum number of common points ?

Let there be two circles of different radii with centers A and C.

There can be four cases with two circles of different radii :

(i) No point of intersection when circles are away from each other.

(ii) One point of intersection when circles touch each other.

(iii) Two points of intersection when circles intersect each other.

(iv) Infinite, when one circle is completely inside the other and centers are concurrent.

Hence, maximum number of common points are infinite.