Question

Z1 and z2 are two complex numbers such that |z1|=|z2| and arg(z1)+arg(z2)=, then z1 is equal

Answer 1

Answer:

Let |z1|=|z2|=r

arg(z1)=θ

Arg(z2)=Φ

z1=re^(iθ)……(1)

z2=re^(iΦ)………(2)

And θ-φ=π

e^(ix)=cosx + isinx

e^(i(π+x))=cos(π+x)+isin(π+x)

e^(i(π+x))=-cosx-isinx=-(cosx+isinx)=-e^(ix)

Here θ=π+Φ

So this reduces to

e^(iθ)=-e^(iφ)

Multiplying by 'r' on both sides

re^(iθ)=-re^(iφ)

From equations (1) and (2)

The above expression becomes

z1=-z2

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