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Question

A(z1),B(z2) and C(z3) are the vertices of an isosceles triangle in anticlockwise direction with origin as in-centre. If AB=AC, then z2,z1 and kz3 will form (where k=|z1|2|z2||z3|)

A
G.P.
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B
A.G.P.
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C
None of these
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D
A.P.
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Solution

The correct option is A G.P.

Let BAC=2θ
AB=AC
B=C=12(π2θ)=π2θ
AIC=π(θ+C2)=(3π4θ2),BIA=(3π4θ2)
z3z1=z3z1e(i3π4θ2) ...(1)
and z2z1=z2z1ei(3π4θ2) ...(2)

On multiplying equation (1) and (2), we get
z2z3z21=|z2||z3||z1|2
z21=kz2z3
Hence, z2,z1 and kz3 are in G.P.

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