Let us a discuss yet one more complex number based IITJEE mains problem.
Problem:
Let and
be nth roots of unity, which subtend a right angle at the origin. Then, integer n must be of the form __________________. (fill in the blank).
Solution:
nth roots of unity are given by
where .
Let
Let where
,
As the join of and
subtends a right angle at the origin, we deduce that
is purely imaginary.
, for some real k
. Thus, n must be of the form 4k.
More later,
Nalin Pithwa