Hello Everyone,
Thank you for your responses. The issue of when the meeting began - whether at 2 pm or when I arrived - is immaterial. All you need to know is that two points in time are related by having the positions of the minute-hand and hour-hand PRECISELY switched.
Here's the solution:
Let MA denote the position of the minute-hand when I arrived, and ME the position of the minute-hand when the meeting ended. Similarly, let HA denote the position of the hour-hand when I arrived, and HE the position of the hour-hand when the meeting ended.
So HA=2+x and MA=12x, for some positive x less than 1. The variable x denotes the fraction of an hour that I was late. When the hour-hand has a certain displacement of x from the position 2, the minute-hand has a displacement of 12x from position zero (zero being same as position 12).
Similarly, HE=3+y since the minute-hand at the time of my arrival was between 3 and 4, which is when the meeting ended. Also ME=12y.
Now, according to the problem, we have HA=ME and MA=HE.
So, we have the system of equations {2+x=12y, 12x=3+y} to solve.
Once the system is solved, then the answer needs to be converted to HH:MM:SS with SS being to the nearest second.
A BONUS QUESTION: How many points in time in a 12-hour period are there that have the property that a precise switch of positions of the hour-hand and minute hand gives rise to 'legitimate' positions of the hands?