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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 find z2 and z3

A
-2 & 1 - i 3
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B
-2 i & 1 - i 3
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C
-2 & 1 + i 3
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D
-2i & 1 + i 3
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Solution

The correct option is A -2 & 1 - i 3
z2=¯¯¯¯¯z1
=13i
Now since they are vertices of an equilateral triangle.
Hence
z21+z22+z23=z1.z2+z2.z3+z3.z1
223i2+23i+z22=4+2z2
4+z22=4+2z2
z222z28=0
z24z+2z8=0
(z4)(z+2)=0
Hence
z=4 and z=2
Hence
The vertices are 2,(13i).

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