Question

Let z1, z2, and z3 be non-zero complex numbers satisfying$z^2=iz`$., where i =√-1 ? What is the value of z1+z2+z3?

note z` means z modulus:
Refer the attachment​

Answer 1

Step-by-step explanation:

We have,

${z}^{2} = iz^{ - }$

$\implies{z}^{3} = i |z|^{2}$

Now, let $\omega = i|z|^{2} .... ( a purely imaginary complex number$

Then,

$\implies {z}^{3} = \omega$

$\implies {z}^{3} - \omega = 0$

$\implies {z}^{3} + 0. {z}^{2} + 0.z - \omega = 0$

Since, $z_{1}, z_{2} and z_{3}$ satisfying the given equation, so they must be the roots of the above equation, so,

$z_{1} + z_{2} + z_{3} = 0$