If `z_(1)`, `z_(2)` are complex numbers such that `Re(z_(1))=|z_(1)-2|`, `Re(z_(2))=|z_(2)-2|` and `arg(z_(1)-z_(2))=pi//3` , then `Im(z_(1)+z_(2))=`

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If `z_(1)`, `z_(2)` are complex numbers such that `Re(z_(1))=|z_(1)-2|`, `Re(z_(2))=|z_(2)-2|` and `arg(z_(1)-z_(2))=pi//3` , then `Im(z_(1)+z_(2))=`

If `z_(1)`, `z_(2)` are complex numbers such that `Re(z_(1))=|z_(1)-2|`, `Re(z_(2))=|z_(2)-2|` and `arg(z_(1)-z_(2))=pi//3` , then `Im(z_(1)+z_(2))=`
A. `2//sqrt(3)`
B. `4//sqrt(3)`
C. `2//sqrt(3)`
D. `sqrt(3)`

Monjur Alahi

4 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Correct Answer - B
`(b)` Let `z_(1)=x_(1)+iy_(1)` and `z_(2)=x_(2)+iy_(2)`
Given `Re(z_(1))=|z_(1)-2|`, `Re(z_(2))=|z_(2)-2|`
`:.y_(1)^(2)-4x_(1)+4=0` and `y_(2)^(2)-4x_(2)+4=0`
So that `(y_(1)-y_(2))/(x_(1)-x_(2))=(4)/(y_(1)+y_(2))` ……………`(i)`
Given `arg(z_(1)-z_(2)=pi//3`
`implies (y_(1)-y_(2))/(x_(1)-x_(2))=sqrt(3)` .........`(ii)`
From `(i)` and `(ii) implies y_(1)+y_(2)=(4)/(sqrt(3))`