Angular speed of planet #2, w2 = 2pi/11 rad/day.
Planet #1, being faster, should have revolved once before planet #2 finishes its first revolution. Or, let us assume that in the same amount of time, the two planets will be alligned again such that the faster one is ahead by one revolution.
Angular distance, or angle, = (angular speed)*(time)
In t days time,
Planet #1 covers (w1)*t
Planet #2 covers (w2)*t
(w1)*t -(w2)*t = 1 revolution = 2pi
(2pi/7)*t -(2pi/11)*t = 2pi
(1/7)t -(1/11)t = 1
(4/77)t = 1
t = 77/4 = 19.25 days
That means, in 19.25 days the two planets will be aligned again.
In 19.25 days,
>>>planet #1, revolving once every 7 days, should have gone (2 +5.25/7) revolutions....or 2.75 revs.
>>>planet #2, at 11 days per rev, should have gone (1 +8.25/11) revolutions ....or 1.75 revs
That means the faster planet has completed more than 2 revs before it was able to catch up with the slower planet. It looks like our assumption is wrong.
But then, the slower planet has completed more that 1 rev when it was caught up. Meaning, between the two planets, there is only 1 rev difference. So our assumption is still correct.
Therefore, after 19 days and 6 hours, the two planets are again aligned. ----answer.