Q4 of 153 Page 14

If z1 and z2 are two complex number such that |z1| = |z2| and arg (z1) + arg (z2) = π, then show that z_1 = - bar z_2

Given:


⇒ |z1|=|z2| and arg(z1)+arg(z2)=


Let us assume arg(z1)=θ


⇒ arg(z2)=-θ


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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