Complex Numbers And Quadratic Equations question 150
Question: Suppose $ z_1,z_2,z_3 $ are the vertices of an equilateral triangle inscribed in the circle $ |z|=2 $ . If $ z_1=1+i\sqrt{3}, $ then values of $ z_3 $ and $ z_2 $ are respectively [IIT 1994]
Options:
A) $ -2,1-i\sqrt{3} $
B) $ 2,1+i\sqrt{3} $
C) $ 1+i\sqrt{3},-2 $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
One of the number must be a conjugate of $ z_1=1+i\sqrt{3}i.e.z_2=1-i\sqrt{3} $ or $ z_3=z_1{e^{i2\pi /3}} $ and $ z_2=z_1{e^{-i2\pi /3}} $ $ z_3=(1+i\sqrt{3})[ \cos ( \frac{2\pi }{3} )+i\sin \frac{2\pi }{3} ]=-2 $ Aliter: Obviously $ |z|=2 $ is a circle with centre $ O(0,0) $ and radius 2. Therefore, $ OA=OB=OC $ and this is satisfied by (a) because two vertices of any triangle cannot be same.
BETA
