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Question

If from a point P representing the complex number z1 on the curve |z|=2, two tangents are drawn from P to the curve |z|=1, meeting at points Q(z2) and R(z3), then

A
Complex number (z1+z2+z3)3 will be on the curve |z|=1
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B
(4¯¯¯z1+1¯¯¯z2+1¯¯¯z3)(4z1+1z2+1z3)=9
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C
arg(z2z3)=2π3
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D
Orthocenter and circumcenter of ΔPQR will concide
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Solution

The correct options are
A Complex number (z1+z2+z3)3 will be on the curve |z|=1
B (4¯¯¯z1+1¯¯¯z2+1¯¯¯z3)(4z1+1z2+1z3)=9
C arg(z2z3)=2π3
D Orthocenter and circumcenter of ΔPQR will concide
Since OQ=1 and OP=2, so sin(OPQ)=1/2 and hence QPR=π/3. Then PQR is equilateral. Also, OMQR. Then from OMQ,OM=1/2. Hence, MN=1/2. Then centroid of PQR lies on |z|=1.
AS PQR is an equilateral triangle, so orthocenter, circumcenter, and centroid will coincide. Now,
z1+z2+z33=1
or |z1+z2+z|2=9
or (z1+z2+z3)(¯¯¯z1+¯¯¯z2+¯¯¯z3)=9
or (4¯¯¯z1+1¯¯¯z2+1¯¯¯z3)(4z1+1z2+1z3)=9 and QOR=120o
170904_117230_ans.png

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