hi shivangi,

in the case of circular motion , when we have to find in how many points the object will meet, what matters is the direction of motion and what is required in this case is the ratio of the speeds in the simplest forms.

when two objects start either in same or in opposite direction then till the first point of meet , the lap difference between them = circumference of the track.(i.e 1 complete lap)

Also, if two objects are moving with the speeds of x and y, then time taken by them to meet at any point in the circular track for the first time is L / (x+y) , if in opposite direction or L / (x-y) if in same direction. Also if the time taken to complete one round by each of them is a sec and b sec respectively then time taken to meet back at the starting point will be LCM (a,b), irrespective of direction of motion.

This boils down to the conclusion that if ratio of speed between two is 3:7,which is in the simplest form  then number of meeting points for moving in same direction will be (7-3 = 4) and if moving in opposite direction is (7+3=10)., means they will meet in either 4 or 10 distinct points based upon the direction of motion and these meeting points always includes the meeting at the starting point