The correct option is E Re(z)=52cos(θ1−θ2)
z1=r1eiθ1, z2=r2eiθ2
z1z2=r1eiθ1r2eiθ2=r1r2ei(θ1−θ2)
Since, 3|z1|=4|z2|
3r1=4r2⇒r1r2=43
z1z2=43ei(θ1−θ2)
Similarly,
z2z1=34ei(θ2−θ1)
Now,
z=3z12z2+2z23z1 =32×43ei(θ1−θ2)+23×34ei(θ2−θ1)
⇒z=2cos(θ1−θ2)+2isin(θ1−θ2) +12cos(θ2−θ1)+12isin(θ2−θ1)
⇒z=52cos(θ1−θ2)+32isin(θ1−θ2)
|z|=√254cos2(θ1−θ2)+94sin2(θ1−θ2)
Im(z)=32sin(θ1−θ2)
Re(z)=52cos(θ1−θ2)
No options matches with the answer,
It is bonus question.