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Question

If a, b, c are the numbers between 0 and 1 such that the points z1=a+i,z2=1+bi, and z3=0 form an equilateral triangle, then a and b is equal to

A
a=2+3,b=2+3
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B
a=23,b=23
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C
a=2+3,b=2+3
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D
a=23,b=23
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Solution

The correct option is D a=23,b=23
Hence
|z3z1|=|z3z2|
a2+1=b2+1
Or
a=±b
and
z21+z22+z23=z1.z2+z3z2+z3z1
a21+2ai+1b2+2bi=(ab)+i(1+ab)
(a2b2)+i(2(a+b))=(ab)+i(1+ab)
Hence
2(a+b)=1+ab
Considering b=a, we get
b2+14b=0
b=4±1642
=2±3
Now
0<a<1 and 0<b<1
Hence
a=b=23

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