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Question

Let A,B,C,D be four concyclic points in order in which AD:AB=CD:CB. If A,B,C are represented by complex numbers a,b,c, then vertex D can be represented as

A
2ac+b(ac)a+c+2b
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B
2ac+b(a+c)a+c+2b
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C
2ac+b(a+c)a+c2b
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D
2acb(a+c)a+c2b
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Solution

The correct option is D 2acb(a+c)a+c2b
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Let complex number representing point D is d and DAB=θ So, BCD=πθ.
Now, applying rotation formula on A and C we get
bada=ABADeiθ and dcbc=CDCBei(πθ)
Multiplying these two, we get
(bada)(dcbc)=AB×CDAD×CBeiπ
d(ba)c(ba)d(bc)a(bc)=1 (ADAB=CDCB)
d=2acb(a+c)a+c2b

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