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Question

Let z1 and z2 be two distinct complex numbers and let z=(1t)z1+tz2 for some real number t with 0<t<1. If arg(ω) denotes the principal argument of a non-zero complex number ω, then

A
|zz1|+|z+z2|=|z1z2|
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B
arg(zz1) = arg(zz2)
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C
zz1¯¯¯z¯¯¯¯¯z1z2z1¯¯¯¯¯z2¯¯¯¯¯z1=0
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D
arg(zz1) = arg(z2z1)
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Solution

The correct options are
A |zz1|+|z+z2|=|z1z2|
C zz1¯¯¯z¯¯¯¯¯z1z2z1¯¯¯¯¯z2¯¯¯¯¯z1=0
D arg(zz1) = arg(z2z1)
Given, z=(1t)z1+tz2(1t)t

Clearly, z divides z1 and z2 in the ratio of t : (1-t), 0<t<1
AP+BP=ABi.e.|zz1|+|zz2|=|z1z2| Option (a) is true.
And arg(zz1) =arg(z2z)
=arg(z2z1)
Option (b) is false and option (d) is true.
Also, arg(zz1)=arg(z2z1)
arg(zz1z2x1)=0
zz1z2z1 is purely real.
zz1z2z1=¯z¯¯¯¯z1¯¯¯¯z2¯¯¯¯z1
or zz1¯¯¯z¯¯¯¯¯z1z2z1¯¯¯¯¯z2¯¯¯¯¯z1=0
Option (c ) is correct. Hence, ( a, c, d ) is the correct option.

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