Complex Numbers And Quadratic Equations question 636
Question: Let $ A_0A_1A_2A_3A_4A_5 $ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $ A_0A_1,A_0A_2 $ and $ A_0A_4 $ is
Options:
A) $ \frac{3}{4} $
B) $ 3\sqrt{3} $
C) 3
D) $ \frac{3\sqrt{3}}{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
Let the vertices be $ z_0,z_1,……..z_5 $ w.r.t centre O as origin $ |z_0|=1, $ $ A_0A_1=|z_1-z_0|=|z_0{e^{i\theta }}-z_0| $
$ \therefore A_0A_1=|z_0||cos\theta +isin\theta -1| $ $ =1.\sqrt{{{(\cos \theta -1)}^{2}}+{{\sin }^{2}}\theta }=\sqrt{2(1-\cos \theta )} $
$ \therefore A_0A_1=\sqrt{2.2{{\sin }^{2}}\frac{\theta }{2}}=2\sin \frac{\theta }{2} $ Where $ \theta =\frac{2\pi }{6}=\frac{\pi }{3}. $ Replacing $ \theta $ by $ 2\theta $ and $ 4\theta $ we get, $ A_0A_2=2\sin \frac{2\theta }{2}=2\sin \theta $ & $ A_0A_4=2\sin \frac{4\theta }{2}=2\sin 2\theta $
$ \therefore (A_0A_1)(A_0A_2)(A_0A_4) $ $ =8\sin \frac{\pi }{6}\sin \frac{\pi }{3}\sin \frac{2\pi }{3} $ $ =8( \frac{1}{2} )( \frac{\sqrt{3}}{2} )( \frac{\sqrt{3}}{2} )=3 $
BETA
