Q. 44.3( 7 Votes )

If z1 and z2 are two complex number such that |z1| = |z2| and arg (z1) + arg (z2) = π, then show that

Answer :

Given:


|z1|=|z2| and arg(z1)+arg(z2)=


Let us assume arg(z1)=θ


arg(z2)=


We know that z=|z|(cosθ+isinθ)


z1=|z1|(cosθ+isinθ)-----------------(1)


z2=|z2|(cos(-θ)+isin(-θ))


z2=|z2|(-cosθ+isinθ)


z2=-|z2|(cosθ-isinθ)


Now we find the conjugate of z2


=-|z2|(cosθ+isinθ) ( )


Now,



( |z1|=|z2|)


z1=-


Thus proved.


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