At what time between 10 o'clock and 11 o'clock are the two hands of a clock symmetric with respect to the vertical line (give the answer to the nearest second)?

Symmetry with respect to the vertical line (the 12-6 axis) means the sum of the angles of the hands from the 12 o'clock position must be 360 degrees. At 10:x, the hour hand is at 300 + 0.5x degrees and the minute hand is at 6x degrees. Solving 300 + 0.5x + 6x = 360 gives 6.5x = 60, so x = 120/13 minutes, which is approximately 9 minutes and 13.8 seconds.