Statement I Let z1 and z2 be two complex numbers such that |z1+z2|=|z1|+|z2|, then arg (z1)- arg (z2)=0 Statement II arg ((z1/z2))= arg (z1)- arg (z2).

Question

Statement I Let z1 and z2 be two complex numbers such that |z1+z2|=|z1|+|z2|, then arg (z1)- arg (z2)=0 Statement II arg ((z1/z2))= arg (z1)- arg (z2). (A) Statement I is correct (B) Statement II is coprect (C) Both are correct (D) Neither I nor II is correct

Tardigrade Admin · Accepted Answer

Let z1=r1( cos θ1+i sin θ1) and z2=r2( cos θ2+i sin θ2) where r1=|z1|, arg (z1)=θ1, r2=|z2|, arg (z2)=θ2 we have, |z1+z2|=|z1|+|z2| =|r1( cos θ1+ cos θ2)+r2( cos θ2+ sin θ2)|=r1+r2 r12+r22+2 r1 r2 cos (θ1 θ2)=(r1+r2)2 ⇒ cos (θ1-θ2)=1 ⇒ θ1-θ2=0 ⇒ θ1=θ2 text i.e; arg z1= arg z2

Q. Statement I Let z1​ and z2​ be two complex numbers such that ∣z1​+z2​∣=∣z1​∣+∣z2​∣, then arg(z1​)−arg(z2​)=0 Statement II arg(z2​z1​​)=arg(z1​)−arg(z2​).

Statement I is correct

Statement II is coprect

Both are correct

Neither I nor II is correct

Q. Statement I Let $z_{1}$ and $z_{2}$ be two complex numbers such that $∣ z_{1} + z_{2} ∣ = ∣ z_{1} ∣ + ∣ z_{2} ∣$ , then $ar g (z_{1}) - ar g (z_{2}) = 0$
Statement II $ar g (\frac{z _{1}}{z _{2}}) = ar g (z_{1}) - ar g (z_{2})$ .