Let z1 and z2 be two complex numbers such that barz1+i barz2=0 and arg (z1z2)=π, then find arg (z1).

Question

Let z1 and z2 be two complex numbers such that barz1+i barz2=0 and arg (z1z2)=π, then find arg (z1). (A) π (B) π/4 (C) π/2 (D) 3π/4

Tardigrade Admin · Accepted Answer

Given that barz1+i barz2 =0 ⇒ z1=iz2, i.e., z2=-iz1 Thus arg (z1z2)=arg z1+arg z2=arg z1+arg (-iz1)=π ⇒ arg (-iz12)=π ⇒ arg (-i)+arg(z12)=π ⇒ arg (-i)+2 arg (z1)=π ⇒ (-π/2)+2 arg (z1)=π ⇒ arg (z1)=(3π/4)

Q. Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overset{z}{ˉ}_{1} + i \overset{z}{ˉ}_{2} = 0$ and arg $(z_{1} z_{2}) = π$ , then find arg $(z_{1})$ .

$π$

$π /4$

$π /2$

$3 π /4$

Q. Let z1​ and z2​ be two complex numbers such that zˉ1​+izˉ2​=0 and arg (z1​z2​)=π, then find arg (z1​).

π

π/4

π/2

3π/4

Given that zˉ1​+izˉ2​=0 ⇒z1​=iz2​, i.e., z2​=−iz1​ Thus arg (z1​z2​)=argz1​+argz2​=argz1​+arg(−iz1​)=π ⇒arg(−iz12​)=π ⇒arg(−i)+arg(z12​)=π ⇒arg(−i)+2arg(z1​)=π ⇒2−π​+2arg(z1​)=π ⇒arg(z1​)=43π​

Q. Let $z_{1}$ and $z_{2}$ be two complex numbers such that $\overset{z}{ˉ}_{1} + i \overset{z}{ˉ}_{2} = 0$ and arg $(z_{1} z_{2}) = π$ , then find arg $(z_{1})$ .

$π$

$π /4$

$π /2$

$3 π /4$