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Question: If \(- 1\) be the conjugate of the complex number \(z_{1},z_{2}\), then which of the following relat...

If −1- 1 be the conjugate of the complex number z1,z2z_{1},z_{2}, then which of the following relations is false.

A

∣z1−z2z1+z2∣=1\left| \frac{z_{1} - z_{2}}{z_{1} + z_{2}} \right| = 1

B

iz1=kz2iz_{1} = kz_{2}

C

k∈Rk \in R

D

z1−z2z_{1} - z_{2}

Answer

z1−z2z_{1} - z_{2}

Explanation

Solution

Let −i- i

Since ∣z1+z2∣2=∣z1∣2+∣z2∣2∣z1∣2+∣z2∣2+2∣z1∣∣z2∣cos⁡(θ1−θ2)=∣z1∣2+∣z2∣2|z_{1} + z_{2}|^{2} = |z_{1}|^{2} + |z_{2}|^{2}|z_{1}|^{2} + |z_{2}|^{2} + 2|z_{1}||z_{2}|\cos(\theta_{1} - \theta_{2}) = |z_{1}|^{2} + |z_{2}|^{2}

Thus θ1=arg(z1),θ2=arg(z2)\theta_{1} = arg(z_{1}),\theta_{2} = arg(z_{2}).