Complex Numbers And Quadratic Equations question 162
Question: Let $ z_1 $ and $ z_2 $ be nth roots of unity which are ends of a line segment that subtend a right angle at the origin. Then n must be of the form [IIT Screening 2001; Karnataka 2002]
Options:
A) 4k + 1
B) 4k + 2
C) 4k + 3
D) 4k
Show Answer
Answer:
Correct Answer: D
Solution:
$ {1^{1/n}}=\cos \frac{2r\pi }{n}+i\sin \frac{2r\pi }{n} $ Let $ z_1=\cos \frac{2r_1\pi }{n}+i\sin \frac{2r_1\pi }{n} $ and $ z_2=\cos \frac{2r_2\pi }{n}+i\sin \frac{2r_2\pi }{n} $ . Then $ \angle Z_1OZ_2=amp( \frac{z_1}{z_2} )=amp(z_1)-amp(z_2) $ $ =\frac{2(r_1-r_2)\pi }{n}=\frac{\pi }{2} $ (Given) \ $ n=4(r_1-r_2) $ =4 × integer, so n is of the form 4 k.
BETA
