Complex Numbers And Quadratic Equations question 251
Question: The cube roots of unity when represented on the Argand plane form the vertices of an [IIT 1988; Pb. CET 2004]
Options:
A) Equilateral triangle
B) Isosceles triangle
C) Right angled triangle
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
The cube roots of unity are $ 1,\omega ,{{\omega }^{2}} $ . We know that if $ z_1,z_2,z_3 $ are the vertices of an equilateral triangle in the Argand plane. Then $ z_1^{2}+z_2^{2}+z_3^{2}=z_1z_2+z_2z_3+z_3z_1 $ If we take $ z_1=1,z_2=\omega ,z_3={{\omega }^{2}} $ Then $ z_1^{2}+z_2^{2}+z_3^{2}=1+{{\omega }^{2}}+{{\omega }^{4}}=0 $ and $ z_1z_2+z_2z_3+z_3z_1=1.\omega +\omega .{{\omega }^{2}}+{{\omega }^{2}}.1 $ $ =\omega +{{\omega }^{3}}+{{\omega }^{2}}=1+\omega +{{\omega }^{2}}=0 $ Thus $ z_1^{2}+z_2^{2}+z_3^{2}=z_1z_2+z_2z_3+z_3z_1 $ Therefore triangle is equilateral. Note: Students should remember this question as a fact.
BETA
