Time Lines

 

1. The time is 2:12.
2. The time is 9:48.

The hour hand is exactly on a line every 12 minutes, so we only need to look at times ending in 00, 12, 24, 36, and 48.
The hands exactly overlap at 12:00. They are 5 lines apart at 11:00 and 1:00, and farther apart at other hours.
At times ending in 12, the hands are closest at 2:12, where they are 1 line apart.
At times ending in 24, the hands are closest at 4:24, where they are 2 lines apart.
At times ending in 36, the hands are closest at 7:36, where they are 2 lines apart.
At times ending in 48, the hands are closest at 9:48, where they are 1 line apart.