At what time between 9 o'clock and 10 o'clock will the hand of a clock point in opposite direction?

At 9 o'clock, the smaller angle between hour hand and minute hand is 90 degrees.
As time passes, this smaller angle will increase.
For the hands to point in exact opposite directions, the angle has to become 180 degrees.
It means that the minute hand has to gain 90 degrees more over the hour hand.
Every minute, the minute hand gains 6 - 0.5 = 5.5 degrees over the hour hand.
To gain 90 more degrees, the minute hand will take $$\frac{90}{5.5}$$ mins ==> $$16\ \frac{4}{11}$$ mins.
So, the hands will face exact opposite directions at $$16\ \frac{4}{11}$$ mins past 9.