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Question

Suppose z1,z2,z3, are the vertices of an equilateral triangle inscribed in the circle |z|=2. If z1=1+i3 then z2 and z3 are equal to

A
2,1i3
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B
2,1i3
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C
2,1+i3
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D
None of these
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Solution

The correct option is A 2,1i3
A(z1),B(z2),C(z3) lie on |z|=2 whose centre is at O(0,0) and radius 2
z1=1+i3 Hence |z|=2 and Arg(z1)=π3
In turn |z2|=|z3|=2 and Arg(z2)=Arg(z1)+120o=180o
z2=2
Further, Arg(z3)=Arg(z2)+120o=300o
Hence, z3=2[cos(2ππ3)+isin(2ππ3)]
=2[cosπ3isinπ3]=2(12i32)
=13i
Thus, z2=2 and z3=1i3

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