`z_(1) "the"z_(2) "are two complex numbers such that" |z_(1)| = |z_(2)|`. "and" arg `(z_(1)) + arg (z_(2) = pi," then show that "z_(1) = - barz_(2).`

`z_(1) "the"z_(2) "are two complex numbers such that" |z_(1)| = |z_(2)

1.	`z_(1) "the"z_(2) "are two complex numbers such that" \|z_(1)\| = \|z_(2)\|`. "and" arg `(z_(1)) + arg (z_(2) = pi," then show that "z_(1) = - barz_(2).`
Answer» Let `z_(1) = r_(1) (costheta_(1) + i sintheta _(1))` and `z_(1) = r_(1)(costheta_(1) + i sintheta _(1)) are two complex numbers. Given that `\|z_(1)\| = \|z_(2)\|` and `arg (z_(1)) + arg (z_(2)) = pi` If `\|z_(1)\|= \|z_(2)\|` `rArr r_(1) = r_(2) ...(i) and if `arg (z_(1)) + arg (z_(2)) = pi` ` rArr theta _(1) + theta_(2) = pi` `rArr theta ^(1) = pi - theta_(2)` Now, `z_(1) = r_(1) (costheta_(1) + i sintheta _(1))` `rArr z_(1) = r_(2) [(cos(pi -theta_(2)) + i sin (pi- theta_(2))]" "[:.r_(1) = r_(2) and theta_(1) = (pi-theta_(2))]` `rArr `z_(1) = r_(2) (-costheta_(2) + i sin theta_(2))` ltbr. `rArr z_(1) = - r_(2) (costheta_(2) - i sin theta_(2))` `z_(1) = -[r_(2) (costheta_(2) - i sin theta_(2))]` `rArr z_(1) = - barz_(2)" "[:. barz = r_(2)(costheta_(2) -i sintheta _(2))]`