Question

If \( z_1 \) and \( z_2 \) are two of the \( n^{th} \) roots of unity such that the line segment joining them subtends a right angle at the origin, then for a positive integer \( k \), \( n \) takes the form:

• A. \(4k\)
• B. \(4k + 1\)
• C. \(4k + 2\)
• D. \(4k + 3\)

Hint:

For roots of unity, angle between points is multiples of \( \frac{2\pi}{n} \).

Answer 1

The Correct Option is A

Step 1: Roots of unity
The \( n^{th} \) roots of unity are points on the unit circle at angles: \[ \frac{2\pi k}{n}, \quad k=0,1,2,\ldots,n-1 \] Step 2: Condition for right angle
If the chord subtends a right angle at origin, the angle difference between \( z_1 \) and \( z_2 \) is \( \frac{\pi}{2} \). Step 3: Relation between \( n \) and right angle
Angle between roots \( = \frac{2\pi m}{n} = \frac{\pi}{2} \implies \frac{2\pi m}{n} = \frac{\pi}{2} \implies n = 4m \) Thus, \( n \) must be a multiple of 4, i.e., \( n = 4k \).