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Question

If θ1 and θ2 are arguments of two non-zero complex numbers z1 and z2respectively such that 3|z1|=4|z2|. If z=3z12z2+2z23z1 then


A
|z|=52  
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B
Re(z)=52cos(θ1θ2)
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C
|z|=54  
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D
Re(z)=52cos(θ1+θ2)
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Solution

The correct option is A Re(z)=52cos(θ1θ2)
z1=r1eiθ1,    z2=r2eiθ2
z1z2=r1eiθ1r2eiθ2=r1r2ei(θ1θ2)
Let θ1θ2=α
Since, 3|z1|=4|z2|
3r1=4r2r1r2=43
z1z2=43eiα
Similarly,
z2z1=34eiα
Now,
z=3z12z2+2z23z1  =32×43eiα+23×34eiα
z=2cosα+2isinα       +12cos(α)+12isin(α)

z=52cosα+32isinα

|z|=254cos2α+94sin2α
Im(z)=32sinα
Re(z)=52cosα

Mathematics

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