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Question

If z is a complex number satisfying arg(z+a)=π6 and arg(za)=2π3, aR+, then

A
|z|=a
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B
|z|=2a
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C
arg(z)=π2
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D
arg(z)=π3
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Solution

The correct options are
A |z|=a
D arg(z)=π3
Given : arg(z+a)=π6 and arg(za)=2π3

So, z lies on the circle whose diameter is AB
OB=a=OC
Therefore,
COB=π3z=a(cosπ3+isinπ3)

Hence,
|z|=aarg(z)=π3

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