Suppose $$z1$$, $$z2$$ are two distinct complex numbers and a,b are real numbers.$$Statement-1$$: If $$z1+z2=a$$, $$z1z2=b$$, then $$arg(z1z2)=0Statement-2$$: If $$z1+z2=a$$, $$z1z2=b$$, then $$z1$$¯$$=z2$$

Question

Suppose $$z1$$, $$z2$$ are two distinct complex numbers and a,b are real numbers.$$Statement-1$$: If $$z1+z2=a$$, $$z1z2=b$$, then $$arg(z1z2)=0Statement-2$$:  If $$z1+z2=a$$, $$z1z2=b$$, then $$z1$$¯$$=z2$$

Infinity Learn · Accepted Answer

$$Statement-2$$ is false. This can be seen by taking $$z1$$ and $$z2$$ to be distinct real numbers.For $$statement-1$$, take $$z1=$$α$$1+i$$β$$1$$, $$z2=$$α$$2+i$$β$$2$$. We have     β$$1+$$β$$2=0$$,α$$1$$β$$2+$$α$$2$$β$$1=0If$$ β$$2=0$$, then β$$1=0$$, and $$statement-1$$ is true.If β$$2$$≠$$0$$, then β$$1=-$$β$$2$$ and α$$1-$$α$$2$$β$$2=0$$⇒   α$$1$$−α$$2=0$$ or α$$1=$$α$$2Thus$$, α$$2+i$$β$$2=$$ α$$1$$−iβ$$1$$  ⇒ $$z2=z1$$¯In this case also arg⁡$$z1z2=arg$$⁡$$z1z$$¯$$1=arg$$⁡$$z12=0$$

Suppose $z_{1}, z_{2}$ are two distinct complex numbers and $a, b$ are real numbers.
Statement $- 1 :$ If $z_{1} + z_{2} = a, z_{1} z_{2} = b,$ then $a r g (z_{1} z_{2}) = 0$
Statement $- 2 :$ If $z_{1} + z_{2} = a, z_{1} z_{2} = b,$ then $\bar{z_{1}} = z_{2}$

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Suppose z1, z2 are two distinct complex numbers and a,b are real numbers.Statement-1: If z1+z2=a, z1z2=b, then arg(z1z2)=0Statement-2: If z1+z2=a, z1z2=b, then z1¯=z2

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Suppose $z_{1}, z_{2}$ are two distinct complex numbers and $a, b$ are real numbers.
Statement $- 1 :$ If $z_{1} + z_{2} = a, z_{1} z_{2} = b,$ then $a r g (z_{1} z_{2}) = 0$
Statement $- 2 :$ If $z_{1} + z_{2} = a, z_{1} z_{2} = b,$ then $\bar{z_{1}} = z_{2}$