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