Complex Numbers And Quadratic Equations question 323
Question: The centre of a regular polygon of $ n $ sides is located at the point $ z=0 $ and one of its vertex $ z_1 $ is known. If $ z_2 $ be the vertex adjacent to $ z_1 $ , then $ z_2 $ is equal to
Options:
A) $ z_1( \cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n} ) $
B) $ z_1( \cos \frac{\pi }{n}\pm i\sin \frac{\pi }{n} ) $
C) $ z_1( \cos \frac{\pi }{2n}\pm i\sin \frac{\pi }{2n} ) $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let $ A $ be the vertex with affix $ z_1 $ . There are two possibilities of $ z_2, $ i.e., $ z_2 $ can be obtained by rotating $ z_1 $ through $ \frac{2\pi }{n} $ either in clockwise or in anticlockwise direction. \ $ \frac{z_2}{z_1}=| \frac{z_2}{z_1} |{e^{\pm \frac{i2\pi }{n}}} $
Þ $ z_2=z_1{e^{\pm \frac{i2\pi }{n}}} $ , $ (\because |z_2|=|z_1|) $
Þ $ z_2=z_1( \cos \frac{2\pi }{n}\pm i\sin \frac{2\pi }{n} ) $
BETA
