Q. 164.5( 6 Votes )

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

Answer :

Let z1 = |z1| (cos θ1 + I sin θ1) and z2 = |z2| (cos θ2 + I sin θ2)


Given that |z1| = |z2|


And arg (z1) + arg (z2) = π


θ1 + θ2 = π


θ1 = π – θ2


Now, z1 = |z2| (cos (π - θ2) + I sin (π - θ2))


z1 = |z2| (-cos θ2 + I sin θ2)


z1 = -|z2| (cos θ2 – I sin θ2)


z1 = - [|z2| (cos θ2 – I sin θ2)]


z1 = -z̅ 2


Hence proved.


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