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Question

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

A
|zz1|+|zz2|=|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|+|zz2|=|z1z2|
C zz1¯¯¯z¯¯¯z1z2z1¯¯¯z2¯¯¯z1=0
D Arg(zz1)=Arg(z2z1)
Given z=(1t)z1+tz2
zz1z2z1=targ(zz1z2z1)=0 (1) [As t is real hence the zz1z2z1 has arg is 0 ]

arg(zz1)=arg(z2z1)

zz1z2z1=¯¯¯z¯¯¯¯¯z1¯¯¯¯¯z2¯¯¯¯¯z1 As [t is real then t and its conjugate are equal]
zz1¯¯¯z¯¯¯z1z2z1¯¯¯z2¯¯¯z1=0

AP+PB=AB
|zz1|+|zz2|=|z1z2|.

103892_32016_ans.PNG

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