At what time, between 10 o’clock and 11 o’clock are two hands of the clock symmetric with respect to the vertical line (give the answer to the nearest second)?A. 10 h 9 min 13 secB. 10 h 9 min 14 secC. 10 h 9 min 22 secD. 10 h 9 min 50 sec

To solve this problem, we need to determine the time when the two hands of the clock are symmetric with respect to the vertical line, between 10 o'clock and 11 o'clock.

Step 1: Understanding the Symmetry Condition
The vertical line divides the clock into two halves. The hands of the clock will be symmetric if they are at equal angles from the vertical line, but in opposite directions. The minute hand and the hour hand are initially at positions that we need to adjust for symmetry.

Step 2: Determine the Positions of the Hands
At 10:00:

The hour hand is at the 10-hour mark. Since the clock is divided into 12 hours, each hour represents a 30-degree angle (360 degrees / 12 hours). Therefore, the hour hand is at 10 * 30 = 300 degrees from the 12 o'clock position.
The minute hand is at the 12-hour mark, which is at 0 degrees.
Step 3: Calculate the Positions After Some Time
Let the time after 10:00 be t minutes and s seconds.

The minute hand moves 360 degrees in 60 minutes, so it moves 6 degrees per minute (360 degrees / 60 minutes).
The hour hand moves 30 degrees in 60 minutes, so it moves 0.5 degrees per minute (30 degrees / 60 minutes).
Thus, after t minutes, the position of the minute hand will be: Position of minute hand = 6 * t degrees.

The position of the hour hand after t minutes is: Position of hour hand = 300 + 0.5 * t degrees.

Step 4: Symmetry Condition
For the hands to be symmetric with respect to the vertical line, the angle between the hour hand and the vertical line (the 12 o'clock position) should be equal to the angle between the minute hand and the vertical line. The condition of symmetry can be written as:

Angle of hour hand = 360 - Angle of minute hand.

Substitute the expressions for the angles:

300 + 0.5 * t = 360 - 6 * t.

Step 5: Solve for t
Simplify the equation:

300 + 0.5 * t = 360 - 6 * t
300 + 0.5 * t + 6 * t = 360
300 + 6.5 * t = 360
6.5 * t = 60
t = 60 / 6.5
t ≈ 9.23 minutes.

Step 6: Convert to Minutes and Seconds
t ≈ 9.23 minutes = 9 minutes and 0.23 * 60 seconds ≈ 9 minutes and 14 seconds.

Final Answer:
The time when the two hands of the clock are symmetric with respect to the vertical line is approximately 10 h 9 min 14 sec.

Thus, the correct answer is B. 10 h 9 min 14 sec.