Question

what is the relative periodic time when hour and minute hand coincides

Answer 1

I think by "relative periodic time" ,you mean to ask how much time would it take for hour and minute hand to coincide again if they were coinciding initially.

Time taken by minute hand for 1 complete rotation = 1 hour
Time taken by hour hand for 1 complete rotation = 12 hours
So minute hand "completes" 1 rotation per hour
And hour hand "completes" 1/12 rotation per hour

Now since they were coinciding initially, when do you think they will again coincide ?
Since minute hand will lead hour hand, they will coincide next time when minute hand makes just on 1 rotation more than hour hand (just think about it) !
Let time taken for this be "t hours"
In t hours rotation made by minute hand = 1*t =t rotations
And rotations made by hour hand = t * 1/12

According to condition

t-t/12 = 1
This will give t =12/11 hours (approx 66 minutes)

This is the time taken

How many times do you think the hands will coincide in 24 hours(1 day) ?