Let z1 and z2 be two complex numbers such that z1≠ z2 and |z1|≠| z2| . If z1 has a positive real part and z2 has negative imaginary part, then (z1+z2/z1+z2) may be

Question

Let z1 and z2 be two complex numbers such that z1≠ z2 and |z1|≠| z2| . If z1 has a positive real part and z2 has negative imaginary part, then (z1+z2/z1+z2) may be (A) zero or purely imaginary (B) real and positive (C) real and negative (D) none of these

Tardigrade Admin · Accepted Answer

Let z1=cos θ+i sin θ and z2=cos φ+i sin φ ∴ (z1-z2/z1-z2) =(cos θ+i sin θ+cos φ+i sin φ/cos θ+i sin θ-cos φ-i sin φ) =((cos θ+ +cos φ)+i(sin θ+sin φ)/(cos θ-cos φ)+i(sin θ-sin φ)) =-i cot (θ-φ/2) which is purely imaginary if θ ≠φ and zero if (θ-φ/2)=(π/2) or θ=π+φ

Q. Let z1​ and z2​ be two complex numbers such that z1​=z2​ and ∣z1​∣=∣z2​∣ . If z1​ has a positive real part and z2​ has negative imaginary part, then z1​+z2​z1​+z2​​ may be

zero or purely imaginary

real and positive

real and negative

none of these

Q. Let $z_{1}$ and $z_{2}$ be two complex numbers such that $z_{1} \neq = z_{2}$ and $∣ z_{1} ∣ \neq = ∣ z_{2} ∣$ . If $z_{1}$ has a positive real part and $z_{2}$ has negative imaginary part, then $\frac{z _{1} + z _{2}}{z _{1} + z _{2}}$ may be